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ENGLISH GRAMMAR. 



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Clark's English Grammar 1 00 

Clark's Key to English Grammar ... 60 
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Clark's Grammatical Chart 4 oj 

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ful aid, and diverts the pupil by taxing his ingenuity. Teachers who are 
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It the most interesting study of the school course. 

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ETYMOLOGY. 



Smith's Complete Etvmology, l 26 

Containing the Anglo-Saxon, French, Dutch, German, Welsh, Danish, 
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The Topical Lexicon, 1 50 

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V %. 






7- .f .(/?*- — <* 



LOGIC AND UTILITY 



OF 



MATHEMATICS, 



WITH THE BEST METHODS OF INSTRUCTION EXPLAINED 
AND ILLUSTRATED 



BY CHARLES DAVIES, LL.D. 



NEW YOEK: 

PUBLISHED BY A. S. BARNES & Co., 

Ill & 113 WILLIAM STREET. 

1869. 



t • - • -. *** * J. + .^ 



Jt ■*> 




lfotarad according to Act of Congress, in the year Eighteen Hundred and fifty 

By CHARLES DAVIES, 

In the Clerk's Office of the District Court of the United States for the Southern D. ; *4ri 

of New York. 



8TIRI0TTPID BY 

RICHARD C. VALENTIN K, 
"W«w Tom. 



I 

^4 



PREFACE 



The following work is not a series of speculations. It is but 
an analysis of that system of mathematical instruction which 
has been steadily pursued at the Military Academy over a 
quarter of a century, and which has given to that institution 
its celebrity as a school of mathematical science. 

It is of the essence of that system that a principle be taught 
before it is applied to practice ; that general principles and gen- 
eral laws be taught, for their contemplation is far more improving 
to the mind than the examination of isolated propositions ; and 
that when such principles and such laws are fully compre- 
hended, their applications be then taught as consequences or 
practical results. 

This view of education led, at an early day, to the union of 
the French and English systems of mathematics. By this 
union the exact and beautiful methods of generalization, which 
distinguish the French school, were blended with the practical 
methods of the English system. 

The fruits of this new system of instruction have been abun- 
dant. The graduates of the Military Academy have been 
sought for wherever science of the highest grade has been 



PREFACE. 



needed. Russia has sought them to construct her railroads ;* 
the Coast Survey needed their aid ; the works of internal im- 
provement of the first class in our country, have mostly been 
conducted under their direction ; and the recent war with Mexico 
afforded ample opportunity for showing the thousand ways in 
which science — the highest class of knowledge — may be made 
available in practice. 

All these results are due to the system of instruction. In 
that system Mathematics is the basis — Science precedes Art — 
Theory goes before Practice — the general formula embraces all 
the particulars. 

It was deemed necessary to the full development of the plan 
of the work, to give a general view of the subject of Logic. 
The materials of Book I. have been drawn, mainly, from the 
works of Archbishop Whately and Mr. Mill. Although the 
general outline of the subject has but little resemblance to the 
work of either author, yet very much has been taken from both ; 
and in all cases where it could be done consistently with my own 
plan, I have adopted their exact language. This remark is par- 
ticularly applicable to Chapter III., Book I., which is taken, 
with few alterations, from Whately. 

For a full account of the objects and plan of the work, the 
reader is referred to the Introduction. 

Fishkill Landing, 
June, 1850. 

* Major Whistler, the engineer, to whom was intrusted the great enterprise 
of constructing a railroad from St. Petersburg to Moscow, and Maj or Brown, 
who succeeded him at his death, were both graduates of the Military Acad- 
emy. 



CONTENTS. 



INTRODUCTION. 



PA.OI 

Objects and Plan of the Work ........*••>. 11 



BOOK I . 
LOGIC. 

CHAPTER L 
Definitions — Operations of the Mind — Terms defined . . 27 

SECTION 

Definitions 1 — 6 

Operations of the Mind concerned in Reasoning 6 — 12 

Abstraction 1 2 — 14 

Generalization 14 — 22 

Terms — Singular Terms — Common Terms 15 

Classification 16 — 20 

Nature of Common Terms 20 

Science 21 

Art 22 



6 CONTENTS. 



CHAPTER II. 

rxam 
Sources and Means of Knowledge — Induction 41 

SECTION 

Knowledge 23 

Facts and Truths 24—27 

Intuitive Truths 27 

Logical Truths 28 

Logic 29 

Induction 30—34 



CHAPTER III. 

Deduction — Nature of the Syllogism — Its Uses and Ap- 
plications Page 54 

«CTION 

Deduction 34 

Propositions 35 — 40 

Syllogism 40—42 

Analytical Outline of Deduction 42 — 67 

Aristotle's Dictum 54 — 61 

Distribution and Non-distribution of Terms 61 — 67 

Rules for examining Syllogisms 67 

Of Fallacies 68—71 

Concluding Remarks 71 — 75 



CONTENTS. 



BOOK II. 
MATHEMATICAL SCIENCE. 

CHAPTER I. 

Quantity and Mathematical Science defined — Differ- 
ent kinds of Quantity — Language of Mathematics 
explained — Subjects Classified — L t nit of Measure 
defined — Mathematics a Deductive Science . . . .Page 99 

SECTION 

Quantity. 75—79 

Number 79—81 

Space 81—87 

Analysis 87—91 

Language of Mathematics 91 — 94 

Quantity Measured , 94 — 97 

Pure Mathematics 97-101 

Comparison of Quantities 101 

Axioms or Formulas for inferring Equality 102 

Axioms or Formulas for inferring Inequality 102 

CHAPTER IT. 

PAOX 

Arithmetic — Science and Art of Numbers 117 



SECTION I. 

SECTION 

First Notions of Numbers 104 — 107 

Ideas of Numbers Generalized 107 — 110 

Unity and a Unit Defined 110 

Simple and Denominate Numbers 11 1 — 1 1 3 

Alphabet — Words — Grammar 113 

Arithmetical Alphabet 114 

Spelling and Reading in Addition 115 — 120 

Spelling and Reading in Subtraction 120 — 122 



CONTENTS. 



SECTION 

Spelling and Reading in Multiplication 122 

Spelling and Reading in Division 123 

Units increasing by the Scale of Tens 124 — 131 

Units increasing by Varying Scales 131 

Integer Units of Arithmetic 132 

Abstract or Simple Units 132—134 

Units of Currency 1 34—1 36 

Units of Weight 136—139 

Units of Measure 139—150 

Advantages of the System of Unities 150 

System of Unities applied to the Four Ground Rules. 151 — 155 

SECTION II. . 

Fractional Units changing by the Scale of Tens 155 — 158 

Fractional Units in general 158 — 161 

Advantages of the System of Fractional Units 161 — 163 

SECTION III. 

Proportion and Ratio 163—172 

SECTION IV. 

Applications of the Science of Arithmetic 172 — 180 

SECTION V. 

Methods of teaching Arithmetic considered 180 

Order of the Subjects 180—183 

1st. Integer Units 183—185 

2d. Fractional Units 185 

3d. Comparison of Numbers, or Rule of Three 186 — 188 

4th. Practical Part, or Applications of Arithmetic .... 188 

Objections to Classification answered 189 — 191 

Objections to the new Method 191 

Arithmetical Language 192 — 200 

Necessity of exact Definitions and Terms 200 — 206 

How should the Subjects be presented 206 — 209 

Text-Books 209—214 

First Arithmetic 214—227 



CONTENTS. 9 



SECTION 

Second Arithmetic 227-— 231 

Third Arithmetic 231—236 

Concluding Remarks 236 

CHAPTER III. 

Geometry defined — Things of which it treats — Com- 
parison and Properties of Figures — Demonstration 
— Proportion — Suggestions for Teaching .... Page 223 

SECTION 

Geometry 237 

Things of which it treats 238—248 

Comparison of Figures with Units of Measure 249 — 256 

Properties of Figures 256 

Marks of what may be proved 257 

Demonstration 258 — 207 

Proportion of Figures 267 — 270 

Comparison of Figures 270 — 273 

Recapitulation — Suggestions for Teachers 273 

CHAPTER IV. 

Analysis — Algebra — Analytical Geometry — Differen- 
tial and Integral Calculus Page 261 

SECTION 

Analysis 274 — 280 

Algebra 280 

Analytical Geometry 281 — 283 

Differential and Integral Calculus 283 — 286 

Algebra further considered 286 — 296 

Minus Sign 296 — 298 

Subtraction 298 

Multiplication 299—302 

Zero and Infinity « 302—307 

Of the Equation 307—311 

Axioms 311 

Equality — its meaning in Geometry 312 

Suggestions for those who teach Algebra ,. 315 



10 CONTENTS. 



BOOK III, 
UTILITY OF MATHEMATICS. 

CHAPTER I. 

The Utility of Mathematics considered as a Means of 
Intellectual Training and Culture Page ( J!)3 

CHAPTER II. 

The Utility of Mathematics regarded as a Means of 
Acquiring Knowledge — Baconian Philosophy 308 

CHAPTER III. 

The Utility of Mathematics considerep as furnishing 
those Rules of Art which make Knowledge Practi- 
cally Effective 325 



APPENDIX. 

A Course of Mathematics — What it should be 341 

Alphabetical Index 353 



INTRODUCTION 



OBJECTS AND PLAN OF THE WORK. 

Utility and Progress are the two leading utility 
ideas of the present age. They were manifested v ro ~ re9S . 
in the formation of our political and social insti- Their infla- 

ence in ^oV 

tutions, and have been further developed in the ermneul: 
extension of those institutions, with their subdu- 
ing and civilizing influences, over the fairest por- 
tions of a great continent. They are now be- 
coming the controlling elements in our systems i» education. 
of public instruction. 



What, then, must be the basis of that system w^t 

the basis of 

of education which shall embrace within its ho- utility and 



rizon a Utility as comprehensive and a Progress 
as permanent as the ordinations of Providence, 
exhibited in the laws of nature, as made known 
by science ? It must obviously be laid in the 
examination and analysis of those laws ; and 



Progress. 



12 INTRODUCTION. 



Preparatory primarily, in those preparatory studies which fit 
and qualify the mind for such Divine Contem- 
plations. 

Bacons When Bacon had analyzed the philosophy of 

Philosophy. 

the ancients, he found it speculative. The great 
highways of life had been deserted. Nature, 
spread out to the intelligence of man, in all the 
minuteness and generality of its laws — in all the 
harmony and beauty which those laws develop — 
had scarcely been consulted by the ancient phi- 
Phiioso- losophers. They had looked within, and not 

phy of the 

Ancients, without. They sought to rear systems on the 
uncertain foundations of human hypothesis and 
speculation, instead of resting them on the im- 
mutable laws of Providence, as manifested in 
the materia] world. Bacon broke the bars oi 
this mental prison-house: bade the mind go free. 
and investigate nature. 



Foundations Bacon laid the foundations of his philosophy m 
of Bacons organ j c i aws an( j explained the several processes 

Philosophy : © r l 

of experience, observation, experiment, and in- 
duction, by which these laws are made known, 
why op- He rejected the reasonings of Aristotle because 

pcsedto Aris- 

totie's. they were not progressive and useful ; because 
they added little to knowledge, and contributed 
nothing to ameliorate the sufferings and elevate 
the condition of humanity. 



PLAN OF THE WORK. 13 

The time seems now to be at hand when the Practical 
philosophy of Bacon is to find its full develop- 
ment. The only fear is, that in passing from a 
speculative to a practical philosophy, we may, 
for a time, lose sight of the fact, that Practice 

without Science is Empiricism; and that all its true na- 
ture, 
which is truly great in the practical must be the 

application and result of an antecedent ideal. 

What, then, are the sources of that Utility, Whal i& 

the true sys» 

and the basis of that Practical, which the pres- temofedu- 
ent generation desire, and aftei which they are 
so anxiously seeking ? What system of training 
and discipline will best develop and steady the 
intellect of the young ; give vigor and expan- 
sion to thought, and stability to action? What which will 

r r develo P ^d 

course of study will most enlarge the sphere of steady the 
investigation ; give the greatest freedom to the 
mind without licentiousness, and the greatest 
freedom to action consistent with the laws of 
nature, and the obligations of the social com- 
pact ? What subject of study is, from its na- what are 

,., , , . . . the subjects 

ture, most likely to ensure this training, and of study? 
contribute to such results, and at the same time 
lay the foundations of all that is truly great in 
the Practical ? It has seemed to me that math- Mathematics 
ematical science may lay claim to this pre-emi- 
nence. 



14 



INTRODUCTION. 



Nature. 



Founda- The first impressions which the child receives 

ematicai °f Number and Quantity are the foundations of 

knowledge. his mat h em atical knowledge. They form, as it 

were, a part of his intellectual being. The laws 

Laws of f Nature are merely truths or generalized facts, 

in regard to matter, derived by induction from 

experience, observation, and experiment. The 

laws of mathematical science are generalized 

truths derived from the consideration of Number 

and Space. All the processes of inquiry and 

investigation are conducted according to fixed 

laws, and form a science ; and every new thought 

and higher impression form additional links in 

the lengthening chain. 



Number 

and 
Space. 



Mathemat- 
ical knowl- 
edge : 



What it 
does. 



The knowledge which mathematical science 
imparts to the mind is deep — profound — abiding. 
It gives rise to trains of thought, which are born 
in the pure ideal, and fed and nurtured by an 
acquaintance with physical nature in all its mi- 
nuteness and in all its grandeur : which survey 
the laws of elementary organization, by the mi- 
croscope, and weigh the spheres in the balance 
of universal gravitation. 



what ^h e P rocesses of mathematical science serve 

the processes to gj ve men t a ] unity and wholeness. They im- 
part that knowledge which applies the means of 



effect. 



PLAN OF THE WORK. 15 

crystallization to a chaos of scattered particulars, Right knowi- 
and discovers at once the general law, if there themeans^of 
be one, which forms a connecting link between "Jfjj" 8 ** 
them. Such results can only be attained by 
minds highly disciplined by scientific combina- 
tions. In all these processes no fact of science 
is forgotten or lost. They are all engraved on 
the great tablet of universal truth, there to be 
read by succeeding generations so long as the it records 

and preserve! 

laws of mind remain unchanged. This is stri- truth, 
kingly illustrated by the fact, that any diligent 
student of a college may now read the works of 
Newton, or the Mecanique Celeste of La Place. 

The educator regards mathematical science How lh(J 

educator re- 

as the great means of accomplishing his work, gardsmath- 
The definitions present clear and separate ideas, 
which the mind readily apprehends. The axioms The nmmi 
are the simplest exercises of the reasoning fac- 
ulty, and afford the most satisfactory results in 
the early use and employment of that faculty. 
The trains of reasoning which follow are com- 
binations, according to logical rules, of what 
has been previously fully comprehended ; and influence of 

, . , , , .the study of 

the mind and the argument grow together, so mathematics 
that the thread of science and the warp of the on the miQdi 
intellect entwine themselves, and become insep- 
arable. Such a training will lay the foundations 



1G INTRODUCTION. 



of systematic knowledge, so greatly preferable 
to conjectural judgments. 

How the The philosopher regards mathematical science 
regards as the mere tools of his higher vocation. Look- 

mathematics: ^ ^^ & gtea( J y an( J anx i ous e ye tO NgtW*, 

and the great laws which regulate and govern 
all things, he becomes earnestly intent on their 
examination, and absorbed in the wonderful har- 
monies which he discovers. Urged forward by 
its necessity these high impulses, he sometimes neglects that 

to him. . ... 

thorough preparation, in mathematical science, 
necessary to success ; and is not unfrequently 
obliged, like Antaeus, to touch again his mother 
earth, in order to renew his strength. 



The views The mere practical man regards with favor 

of the practi- , i • i 

cai man. only the results of science, deeming the reason- 
ings through which these results are arrived at, 
quite superfluous. Such should remember that 

instruments the mind requires instruments as well as the 
hands, and that it should be equally trained in 
their combinations and uses. Such is, indeed, 
now the complication of human affairs, that to 
do one thing well, it is necessary to know the 
properties and relations of many things. Every 

Everything thing, whether existing in the abstract or in the 
material world ; whether an element of knowl- 



PLAN OF THE WORK. 17 

edge or a rule of art, has its connections and its to know 

. iii • i i tne law ia to 

law: to understand these connections and that knowtne 

law, is to know the thing. When the principle thm "' 

is clearly apprehended, the practice is easj r . 

With these general views, and under a firm Mathematics 
conviction that mathematical science must be- 
come the great basis of education, I have be- 
stowed much time and labor on its analvsis, as 
a subject of knowledge. I have endeavored to 
present its elements separately, and in their con- flow, 
nections ; to point out and note the mental fac- 
ulties which it calls into exercise ; to show why 
and how it develops those faculties ; and in what 
respect it gives to the whole mental machinery 
greater power and certainty of action than can 
be attained by other studies. To accomplish what was 

deemed no- 

these ends, in the way that seemed to me most cessary. 
desirable, I have divided the work into three 
parts, arranged under the heads of Book I., II.. 
and III 

Book I. treats of Logic, both as a science and Logic 
an art ; that is, it explains the laws which gov- 
ern the reasoning faculty, in the complicated 
processes of argumentation, and lays down the Explanation, 
rules, deduced from those laws, for conducting 
such processes. It being one of the leading 

2 



18 INTRODUCTION. 



For what objects to show that mathematical science is the 

best subject for the development and application 

of the principles of logic ; and, indeed, that the 

science itself is but the application of those prin- 

The necessity ciples to the abstract quantities Number and 

of treating it. 

Space, it appeared indispensable to give, in a 
manner best adapted to my purpose, an out- 
line of the nature of that reasoning by means 
of which all inferred knowledge is acquired. 

Book ii. Book II. treats of Mathematical Science. 

Here I have endeavored to explain the nature of 
Ofwhatit the subjects with which mathematical science is 
conversant; the ideas which arise in examining 
and contemplating those subjects; the language 
employed to express those ideas, and the laws of 
their connection. This, of course, led to a class- 
Manner of ification of the subjects; to an analysis of the 
language used, and an examination of the reason- 
ings employed in the methods of proof. 



treating. 



Book in. Book III. explains and illustrates the Utility of 
Mathematics. Mathematics : First, as a means of mental disci- 
pline and training ; Secondly, as a means of ac- 
quiring knowledge ; and, Thirdly, as furnishing 
those rules of art, which make knowledge prac- 
tically effective. 



PLAN OF THE WORK. 19 



Having thus given the general outlines of the classes of 
work, we will refer to the classes of readers for 
whose use it is designed, and the particular ad- 
vantages and benefits which each class may re- 
ceive from its perusal and study. 

There are four classes of readers, who may, Four classes 
it is supposed, be profited, more or less, by the 
perusal of this work : 

1st. The general reader ; First class. 

2d. Professional men and students ; second. 

3d. Students of mathematics and philosophy ; Third. 

4th. Professional Teachers. Fourth. 

First. The general reader, who reads for im. Advantages 
provement, and desires to acquire knowledge, era i re rder. 
must carefully search out the import of language. 
He must early establish and carefully cultivate 
the habit of noting the connection between ideas connec- 
and their signs, and also the relation of ideas to ^ordtlnT 3 
each other. Such analysis leads to attentive ldeas * 
reading, to clear apprehension, deep reflection, 
and soon to generalization. 

Logic considers the forms in which truth must Logic 
be expressed, and lays down rules for reducing 
all trains of thought to such known forms. This 
habit of analyzing arms us with tests by which its value* 
we separate argument from sophistry — truth from 
falsehood. The application of these principles, 



20 INTRODUCTION. 



in the study in the construction of the mathematical science, 
^atblaacs. where the relation between the sign (or language) 
and the thing signified (or idea expressed), is un- 
mistakable, gives precision and accuracy, leads 
to right arrangement and classification, and thus 
prepares the mind for the reception of general 
knowledge. 

Advantages Secondly. The increase of knowledge carries 

to profession- . . . , . r ^ • r • a i • • 1 

aimen. W1 *" ]t the necessity ol classification. A limited 
number of isolated facts may be remembered, or 
a few simple principles applied, without tracing 
out their connections, or determining the placet 
which they occupy in the science of general 
knowledge. But when these facts and principles 
are greatly multiplied, as they are in the learned 
The reason, professions ; when the labors of preceding gen* 
erations are to be examined, analyzed, compared; 
when new systems are to be formed, combining 
all that is valuable in the past with the stimu- 
lating elements of the present, there is occasion 
for the constant exercise of our highest facul- 
Knowiedso ^ es - Knowledge reduced to order ; that is, 
reduced to i alow } eo v e so classified and arranged as to be 

order is ° ° 

science, easily remembered, readily referred to, and ad- 
vantageously applied, will alone suffice to sift 
the pure metal from the dust of ages, and fashion 
it for present use. Such knowledge is Science. 



PLAN OF THE WORK. 21 

Masses of facts, like masses of matter, are ca- Knowledge 
pable of very minute subdivisions ; and when we duced t0 ita 
know the law of combination, they are readily elemeats - 
divided or reunited. To know the law, in any 
case, is to ascend to the source ; and without 
that knowledge the mind gropes in darkness. 

It has been my aim to present such a view objects of 

tVif* work 

of Logic and Mathematical Science as would 
clearly indicate, to the professional student, and 
even to the general reader, the outlines of these 
subjects. Logic exhibits the general formula Logic and 
applicable to all kinds of argumentation, and 
mathematics is an application of logic to the 
abstract quantities Number and Space. 

When the professional student shall have ex- 
amined the subject, even to the extent to which certainty of 
it is here treated, he will be impressed with the 
clearness, simplicity, certainty, and generality of 
its principles ; and will find no difficulty in ma- 
king them available in classifying the facts, and 
examining the organic laws which characterize 
his particular department of knowledge. 

Thirdly. Mathematical knowledge differs from Matnemati- 
every other kind of knowledge in this : it is, as c ede ^ ' 
it were, a web of connected principles spun out 
from a few abstract ideas, until it has become 
one of the great means of intellectual develop- its extent. 



22 INTRODUCTION. 



ment and of practical utility. And if I am per- 

Necessity mitted to extend the figure, I may add, that the 

attheright° web of the spider, though perfectly simple, if we 

pa( see the end and understand the way in which 

it is put together, is yet too complicated to be 

unravelled, unless we begin at the right point, 

and observe the law of its formation. So with 

mathematical science. It is evolved from a few 

— a very few — elementary and intuitive princi- 

How pies : the law of its evolution is simple but ex- 

mathemati- . , . 

cai science is acting, and to begin at the right place and pro- 
constructed. cee j j n t j ie r jgh t wa y ? j s a ]j ttia.t is necessary to 

make the subject easy, interesting, and useful, 
what has I have endeavored to point out the place of 

been at- 
tempted, beginning, and to indicate the way to the math- 
ematical student. I am aware that he is start- 
ing on a road where the guide-boards resemble 
each other, and where, for the want of careful 
observation, they are often mistaken ; I have 
sought, therefore, to furnish him with the maps 
and guide-books of an old traveller. 
Advantages By explaining with minuteness the subjects 
the whole about which mathematical science is conversant, 
subject. t ^ e w h ] e fj e j(j t b e g 0ne over i s a t once sur- 
veyed: by calling attention to the faculties of 
Advantages the mind which the science brings into exercise, 

of consider- 
ing tlie men- We are better prepared to note the intellectual 

tal faculties : . 1 • 1 . i • 11 

operations which the processes require ; and by 



PLAN OF THE WORK. % 23 



a knowledge of the laws of reasoning, and an ofaknowi- 
acquaintance with the tests of truth, we are en- ia WSOfri . a . 
abled to verify all our results. These means have 
been furnished in the following work, and to 
aid the student in classification and arrangement, 
diagrams have been prepared exhibiting separ- what has 

been done. 

ately and in connection all the principal parts of 
mathematical science. The student, therefore, 
who adopts the system here indicated, will find 
his wav clearlv marked out, and will recognise, Advanta ^ e * 

J ° totbestu- 

from their general resemblance to the descrip- dent, 
tions, all the guide-posts which he meets. He 
will be at no loss to discover the connection 
between the parts of his subject. Beginning 
with first principles and elementary combina- 
tions, and guided by simple laws, he will go for- *vnere 

be begins. 

ward from the exercises of Mental Arithmetic 
to the higher analysis of Mathematical Science 
on an ascent so gentle, and with a progress so Drde r 

of progrtjs* 

steady, as scarcely to note the changes. And 
indeed, why should he ? For all mathematical 
processes are alike in their nature, governed by 
the same laws, exercising the same faculties,, unity of 
and lifting the mind towards the same eminence. 



tbe subject. 



o 



Fourthly. The leading idea, in the construe- Advantages 
tion of the work, has been, to afford substantial prof essionai 
aid to the professional teacher. The nature ot teacher - 



24 INTRODUCTION. 



His duties: his duties — their inherent difficulties — the per- 
Diecourage- plexities which meet him at every step — the want 
difficulties: of sympathy and support in his hours of discour- 
agement — (and they are many) — are circum- 
stances which aw T aken a lively interest in the 
hearts of all who have shared the toils, and been 
themselves laborers in the same vineyard. He 
takes his place in the schoolhouse by the road- 
side, and there, removed from the highways of 
Remoteness life, spends his days in raising the feeble mind 
Htei ' of childhood to strength — in planting aright the 
seeds of knowledge — in curbing the turbulence 
of passion — in eradicating evil and inspiring 
good. The fruits of his labors are seen but 
once in a generation. The boy must grow to 
Fruits of manhood and the girl become a matron before 

his efforts, . . 

whenseeu. he is certain that his labors have not been in 
vain. 

Yet, to the teacher is committed the high trust 
of forming the intellectual, and, to a certain ex- 
tent, the moral development of a people. He 

Theimpor- holds in his hands the keys of knowledge. If 

lance of his a! n -, . . , . . 

labors. the " rst mora l impressions do not spring into 
life at his bidding, he is at the source of the 
stream, and gives direction to the current. Al- 
though himself imprisoned in the schoolhouse, 
his influence and his teachings affect all condi- 
tions of society, and reach over the whole hori- 



PLAN OF THE WORK. 



25 



zon of civilization. He impresses himself on The influence 
the young of the age in which he lives, and 
lives again in the age which succeeds him. 



All good teaching must flow from copious sources of 

good teach- 

knowledge. The shallow fountain cannot emit m g. 
a vigorous stream. In the hope of doing some- 
thing that may be useful to the professional 
teacher, I have attempted a careful and full 



Objects for 
which the 

analysis of mathematical science. I have spread work was 

. undertaken. 

out, in detail, those methods which have been 
carefully examined and subjected to the test of 
long experience. If they are the right meth- principles 
ods, they will serve as standards of teaching; . ,, te ' 

3 J o ' mg, thes 

for, the principles of imparting instruction are 
the same for all branches of knowledge. 



The system which I have indicated is com- 
plete in itself. It lays open to the teacher the 
entire skeleton of the science — exhibits all its 
parts separately and in their connection. It 
explains a course of reasoning simple in itself, 
and applicable not only to every process in 
mathematical science, but to all processes of 
argumentation in every subject of knowledge. 

The teacher who thus combines science with 
art, no longer regards Arithmetic as a mere 
treadmill of mechanical labor, but as a means — 



Sygtea 



What it 
present?. 

What it 
explains. 



Science 
combined 
with art : 



26 INTRODUCTION. 



The advan- and the simplest means — of teaching the art and 
intromit, science of reasoning on quantity — and this is 

the logic of mathematics. If he would accom- 

Resuitsof plish well his work, he must so instruct his 

tion. pupils that they shall apprehend clearly, think 

quickly and correctly, reason justly, and open 

their minds freely to the reception of all know! 

edge. 



book r. 

LOGIC. 



tome 
tions 



C II A PTER I. 

DEFINITION'S — OPERATIONS OF THE MIND TERMS DEFINED. 

DEFINITIONS. 

§ 1. Definition is a metaphorical word, which Definition 

a 

literally signifies " laying down a boundary. " metaphorical 
All definitions are of names, and of names only ; 

but in some definitions, it is clearly apparent, defimti 

that nothing is intended except to explain the ^J^ 

meaning of the word; while in others, besides word3: 

explaining the meaning of the word, it is also ° thin™™ 

implied that there exists, or mav exist, a thins; corres P° nd - 

r J ' ° ingtothe 

corresponding to the word. *«*l 



§ 2. Definitions which do not imply the exist- of definitions 

r . . , . . , , which do 

ence of things corresponding to the words de- not implv 
fined, are those usually found in the Dictionary thiDgs a"*" 

J J sponding 

of one's own language. They explain only the *>wonk. 



28 logic. [book I. 



Th meaning of the word or term, by giving some 
explain equivalent expression which may happen to be 
equivalents, better known. Definitions which imply the ex- 
istence of things corresponding to the words de- 
fined, do more than this. 
Definition F or example: "A triangle is a rectilineal fig- 

of a 

triangle; ure having three sides." This definition does 

what .1 • 

two things : 
it o 

implies. i s t j t explains the meaning of the word tri- 

angle ; and, 

2d. It implies that there exists, or may exist, 
a rectilineal figure having three sides. 



or a § 3. To define a word when the definition is 

definition 

which im- to imply the existenee of a thing, is to select 
P iltence o( h° m *H the properties of the thing those which 
athing. are most s j M1 p] e general, and obvious; and the 
Properties properties must l;e very well known to us before 
known. we can decide which are the fittest for this pur- 
pose. Hence, a thing may have many properties 
besides those which are named in the definition 
a definition of the word which stands for it. This second 
truth# kind of definition is not only the best form of ex- 
pressing certain conceptions, but also contributes 
to the development and support of new truths. 

in § 4. In Mathematics, and indeed, in all strict 

Mathematics m ... 

names imply sciences, names imply the existence ot the things 



CHAP. I.] DLflNITIONS. 29 



which they name; and the definitions of those things 
names express attributes of the things ; so that express 
no correct definition whatever, of any mathe- 
matical term, can be devised, which shall not 
express certain attributes of the thing correspond- 
ing to the name. Every definition of this class Definitions 
is a tacit assumption of some proposition which ofthlsclas3 
is expressed by means of the definition, and propositions. 
which gives to such definition its importance. 



§ 5. All the reasonings in mathematics, which Reasoning 

resting on 
definitions ; 



rest ultimately on definitions, do, in fact, rest 



on the intuitive inference, that things corre- 

rests on 

sponding to the words defined have a conceiv- intuitive 



inferences. 



able existence as subjects of thought, and do or 
may have proximately, an actual existence.* 



* There are four rules which aid us in framing defini- Four rules 
tions. 

1st. The definition must be adequate : that is, neither too i s t rule, 
extended, nor too narrow for the word defined. 

2d. The definition must be in itself plainer than the word 2d rule, 
defined, else it would not explain it. 

3d. The definition should be expressed in a convenient 3d ralQt 

number of appropriate words. 

4th. When the definition implies the existence of a thino- 

■ i - » , • r i 4th rule, 

corresponding to the word defined, the certainty of that 

existence must be intuitive. 



30 logic. [book I. 



OPERATIONS OF THE MIND CONCERNED IN REASONING. 

Three opera- § q There are three operations of the mind 
tions of the which are immediately concerned in reasoning. 
1st. Simple apprehension ; 2d. Judgment ; 
3d. Reasoning or Discourse. 

simple a P - 8 7 - Simple apprehension is the notion (or 

prehension, conception) of an object in the mind, analogous 
to the perception of the senses. It is either 

incompicx. Incomplex or Complex. Incomplex Apprehen- 
sion is of one object, or of several without any 
relation being perceived between them, as of a 

complex, triangle, a square, or a circle: Complex is of 
several with such a relation, as of a triangle 
within a circle, or a circle within a square. 

§ 8. Judgment is the comparing together in 
the mind two of the notions (or ideas) which 

Judgment 

denned, are the objects of apprehension, whether com- 
plex or incomplex, and pronouncing that they 
agree or disagree with each other, or that one 
of them belongs or does not belong to, the other : 
for example : that a right-angled triangle and an 
judgment equilateral triangle belong to the class of figures 
cilher called triangles ; or that a square is not a circle. 
r ™ alV Judgment, therefore, is either Affirmative or Neg- 

negative. ^-^ 



CHAP. I.] 



ABSTRACTION. 



31 



§ 9. Reasoning (or discourse) is the act of Reasoning 
proceeding from certain judgments to another 
founded upon them (or the result of them). 

§ 10. Language affords the signs by which Language 
these operations of the mind are recorded, ex- gig™ op- 
pressed, and communicated. It is also an in- bought: 
strument of thought, and one of the principal also, an 

instrument 

helps in all mental operations; and any imper- of thought, 
fection in the instrument, or in the mode of 
using it, will materially affect any result attained 
through its aid. 



§ 11. Every branch of knowledge has, to a 

Every branch 

certain extent, its own appropriate language ; ofknowiedge 
and for a mind not previously versed in the language, 
meaning and right use of the various words and 
signs which constitute the language, to attempt 
the study of methods of philosophizing, would 
be as absurd as to attempt reading before learn- 
ing the alphabet. 



which 
must be 
learned. 



ABSTRACTION. 



§ 12. The faculty of abstraction is that power 
of the mind which enables us, in contemplating 
any object (or objects), to attend exclusively to 



32 LOGIC. [book I. 

some particular circumstance belonging to it, and 
quite withhold our attention from the rest. Thus, 

in 

contempia- if a person in contemplating a rose should make 
the scent a distinct object of attention, and lay 
aside all thought of the form, color, &c, he 
would draw oJf\ or abstract that particular part ; 

the process . 

of drawing and therefore employ the iaculty of abstraction. 
He would also employ the same faculty in con- 
sidering whiteness, softness, virtue, existence 
entirely separate from particular objects. 

§ 13. The term abstraction, is also used to 
denote the operation of abstracting from one or 

The term J 

Abstraction, more things the particular part under cmisi.ler- 

how used. . . . 

ation ; and likewise to designate the state oj the 
mind when occupied by abstract ideas, lie: 
abstraction is used in three sen 

1st. To denote a faculty or power of the 
mind ; 
a process, 2 d. To denote a process of the mind ; and, 

and a state ■ 

of mind. 3J t denote a state of the mind. 



Abstraction 
denotes 
a faculty, 



GENERALIZATION. 



General iza- 



§ 14. Generalization is the process of con- 
tion— the templating the agreement of several objects in 

process of 

contempia- certain points (that is, abstracting the circum- 

4ir»rr i\\n 



ting the 






unginu 1 j/r 

agreement, stances ol agreement, disregarding the diner- 



CHAP. I.] 



TERMS. 



33 



ences), and giving to all and each of these ob- 
jects a name applicable to them in respect to 
this agreement. For example ; we give the 
name of triangle, to every rectilineal figure hav- 
ing three sides : thus we abstract this property 
from all the others (for, the triangle has three 
angles, may be equilateral, or scalene, or right- 
angled), and name the entire class from the prop- 
erty so abstracted. Generalization therefore 
necessarily implies abstraction ; though abstrac- 
tion does not imply generalization. 



of several 
things. 



Generaliza- 
tion 



implies 

abstraction. 



A term. 



TERMS SINGULAR TERMS COMMON TERMS. 

§ 15. An act of apprehension, expressed in 
language, is called a Term. Proper names, or 
any other terms which denote each but a single 
individual, as " Csesar," " the Hudson," " the 
Conqueror of Pompey," are called Singular singular 

terms. 

Terms. 

On the other hand, those terms which denote 
any individual of a whole class (which are form- 
ed by the process of abstraction and generaliza- 
tion), are called Common or general Terms. For common 

terms. 

example ; quadrilateral is a common term, appli- 
cable to every rectilineal plane figure having 
four sides ; River, to all rivers ; and Conqueror, 
to all conquerors. The individuals for which a 
common term stands, are called its Signijicates. signmcaiea 

3 



species. 



Examples 

in 

clasaifl cation 



34 logic. [book I. 



CLASSIFICATION. 

ossification §16. Common terms afford the means of clas- 
sification ; that is, of the arrangement of objects 
into classes, with reference to some common and 
distinguishing characteristic. A collection, com- 
prehending a number of objects, so arranged, is 
Genus, called a Genus or Species — genus being the 
more extensive term, and often embracing many 
species. 

For example: animal is a genus embracing 
every thing which is endowed with life, the pow- 
er of voluntary motion, and sensation. It has 
many species, such as man, beast, bird, &C, If 
we say of an animal, that it is rational, it be 
longs to the species man, for this is the charac- 
teristic of that species. If we say that it has 
wings, it belongs to the species bird, for this, in 
like manner, is the characteristic of the gpeciefl 
bird. 

A species may likewise be divided into das 

or subspecies ; thus the species man, may he 

divided into the classes, male and female, and 

these classes may be again divided until we reach 

the individuals. 

Principles § 17. Now, it will appear from the principles 

of 

classification, which govern this system of classification, that 



Subspecies 

or 

classes. 



CHAP. 1.] CLASSIFICATION'. 35 

the characteristic of a genus is of a more exten- Genus more 

.«.,., r . extensive 

sive signification, but involves lewer particu- than species, 
lars than that ot a species. In like manner, the 
characteristic of a species is more extensive, but 
less full and complete, than that of a subspecies but le9S ful1 
or class, and the characteristics of these less full complete. 
than that of an individual. 

For example ; if we take as a genus the Quadri- 
laterals of Geometry, of which the characteristic 
is, that they have four sides, then every plane 
rectilineal figure, having four sides, will fall under 
this class. If, then, we divide all quadrilaterals / 2 \ 
into two species, viz. those whose opposite sides, 
taken two and two, are not parallel, and those 
whose opposite sides, taken two and two, are 
parallel, we shall have in the first class, all irreg- 
ular quadrilaterals, including the trapezoid (1 and 
2) ; and in the other, the parallelogram, the rhom- 
bus, the rectangle, and the square (3,4, 5, and 6). 

If, then, we divide the first species into two 
subspecies or classes, we shall have in the one, the 
irregular quadrilaterals (1), and in the other, the 
trapezoids (2) ; and each of these classes, being 
made up of individuals having the same char- 
acteristics, are not susceptible of further division. 
If we divide the second species into two 
classes, arranging those which have oblique an- 
gles in the one, and those which have right 




36 



LOGIC. 



[book 1. 



Species 
and 



angles in the other, we shall have in the first, 
two varieties, viz. the common parallelogram 
and the equilateral parallelogram or rhombus (3 
and 4) ; and in the second, two varieties also, 
viz. the rectangle and the square (5 and 6). 
Now, each of these six figures is a quadri- 

Each indi- . , 7 . 

viduai failing lateral; and hence, possesses the characteristic 

| Jj^V of the genus; and each variety of both species 

an the enjoys all the characteristics of the species to 

characteris- 

which it belongs, together with some other dis- 
tinguishing feature ; and similarly, of all classi- 
fications. 



tics. 



Subaltern 
genus. 



Parallelo- 
gram. 



§ 18. Ill special classifications, it is often n<>t 
necessary to begin with the most general char- 
acteristics; and then the genus with which we 
begin, is in fact but a species of a more extended 
classification, and is called a Subaltern Genus. 

For example ; if we begin with the genus Par- 
allelogram, we shall at once have two species, 
viz. those parallelograms whose angles are oblique 
and those whose angles are right angles ; and in 
each species there will be two varieties, viz. in the 
first, the common parallelogram and the rhom- 
bus ; and in the second, the rectangle and square. 



§ 19. A genus which cannot be considered 

Highest ° 

renus. as a species, that is, which cannot be referred 



'JH AI\ I.] N A TUBE OF CO M M ON TERMS. 37 



to a more extended classification, is called the Highest 

genus. 

highest genus ; and a species which cannot be 

Lowest 

considered as a genus, because it contains only species. 
individuals having the same characteristic, is 
called the lowest species. 



NATURE OF COMMON TERMS. 

$ 20. It should be steadily kept in mind, that 
the " common terms'' employed in classification. , 

r A common 

have not, as the names of individuals have, anv termh j 3 

* no real thing 

real existing thing in nature corresponding to corres P° Qd - 

them ; but that each is merely a name denoting 

a certain inadequate notion which our minds inadequate 

have formed of an individual. But as this name 

does not include any thing wherein that indi- does not 

include any 

vidual differs from others of the same class, it thing in" 
is applicable equally well to all or any of them. ind * iv ^ ual3 
Thus, quadrilateral denotes no real thing, dis- dLffer; 
tinct from each individual, but merely any recti- 
lineal figure of four sides, viewed inadequately ; 
that is, after abstracting and omitting all that 
is peculiar to each individual of the class. Bv 

but is 

riiis means, a common term becomes applicable applicable to 

many 

alike to any one of several individuals, or, taken individuals, 
in the plural, to several individuals together. 

Much needless difficulty has been raised re- „ ^ 

> eedless 

spec ting the results of this process : many hav- ***** 
ing contended, and perhaps more having taken 



38 LOGIC. L B00K *■ 

Difficulty in it for granted, that there must be some really 

the interpre- . . r 

tationof existing thing corresponding to each ot those 
^e^. 11 common terms, and of which such term is the 
name, standing for and representing it. For ex- 
ample ; since there is a really existing thing cor- 
Noone responding to and signified by the proper and 
real thmg s j n ~ u i ar name "iEtna," it has been supposed 

correspond- o i i 

in- to each. ^^ foe common term "Mountain" must have 
some one really existing thing corresponding to 
it, and of course distinct from each individual 
mountain, yet existing in each, since the term, 
being common, is applicable, separately, to every 
one of them. 

The fact is, the notion expressed by a common 

term is merely an inadequate (or incomplete) 

inademiate not i° n °f an individual ; and from the very cir- 

notionpar- cums t ance of its inadequacy, it will apply equally 

signaling we u to an y one f se ~ C ral individuals. For ex- 

the thing. J 

ample; if I omit the mention and the consider- 
ation of every circumstance which distinguishes 
iEtna from any other mountain, I then form a 
notion, that inadequately designates iEtna. This 
« MoimUlill » notion is expressed by the common term " moun- 
13 tain," which does not imply any of the peculiar- 

applicable . 

toa11 ities of the mountain ./Etna, and is equally ap- 
plicable to any one of several individuals. 

In regard to classification, we should also bear 
in mind, that we mav fix, arbitrarily, on the 



mountains. 






CHAP. 1 J SCIENCE. 39 



characteristic which we choose to abstract and May fix on 

. , . attributes 

consider as the basis of our classification, disre- arbitrarily 
garding all the rest : so that the same individual cl ^ m ° c r ation 
may be referred to any of several different spe- 
cies, and the same species to several genera, as 
suits our purpose. 

SCIENCE. 

§ 21. Science, in its popular signification, 
means knowledge.* In a more restricted sense, science 

in its general 

it means knowledge reduced to order; that is, sense. 
knowledge so classified and arranged as to be 

easily remembered, readily referred to, and ad- Haaa 

vantageously applied. In a more strict and signification, 
technical sense, it has another signification. 

" Every thins; in nature, as well in the in- _ , 

J & ' \ tews of 

animate as in the animated world, happens or Kant - 
is done according to rules, though we do not 
always know them. Water falls according to 
the laws of gravitation, and the motion of walk- Generallaw8U 
ing is performed by animals according to rules. 
The fish in the water, the bird in the air, move 
according to rules. There is nowhere any want 
of rule. When we think we find that want, we 

Nowhere 

can only say that, in this case, the rules are un- any want of 

i „ , rule. 

known to us. f 

Assuming that all the phenomena of nature 



* Section 23. f Kant. 



40 logic. [book I 

science are consequences of general and immutable laws, 

a technical we may define Science to be the analysis of 

uense ne . t h ose ] aws ^ — comprehending not only the con- 

i analysis ne cted processes of experiment and reasoning 

ot the laws r r o 

of nature, which make them known to man, but also those 
processes of reasoning which make known their 
individual and concurrent operation in the de- 
velopment of individual phenomena. 

a u t . 

§ 22. Art is the application of knowledge to 
Art, practice. Science is conversant about knowl- 
appiication edge . Art j s fa use or application of knowl- 
science, edge, and is conversant about works. Science 
has knowledge for its object : Art has knowledge 
for its guide. A principle of science, when ap- 
plied, becomes a rule of art. The developments 
of science increase knowledge: the applications 
iiid of art add to works. Art, necessarily, presup- 
presupposus p 0ses knowledge : art, in any but its infant state, 

knowledge. r J 

presupposes scientific knowledge ; and if every 
art does not bear the name of the science on 
which it rests, it is only because several sciences 
are often necessary to form the groundwork oi 
a single art. Such is the complication of hu- 
man affairs, that to enable one thing to be done, 
it is often requisite to know the nature and prop 



Many things 

must be 
known be- 
fore one can 

be done, erties of many things. 



CHAP. II.] 



KNOWLEDGE. 



41 



CHAPTER II. 



SOURCES AND MEANS OF KNOWLEDGE — INDUCTION 



KNOWLEGDE. 

§ 23. Knowledge is a clear and certain con- 
ception of that which is true, and implies three 
things : 

1st. Firm belief; 2d. Of what is true; and, 
3d. On sufficient grounds. 

If any one, for example, is in doubt respecting 
one of Legendre's Demonstrations, he cannot 
be said to know the proposition proved by it. If, 
again, he is fully convinced of any thing that is 
not true, he is mistaken in supposing himself to 
know it; and lastly, if two persons are each fully 
confident, one that the moon is inhabited, and 
the other that it is not (though one of these 
opinions must be true), neither of them could 
properly be said to know the truth, since he 
cannot have sufficient proof of. it. 



Knowledge 
a clear con- 
ception of 
what is true 

Implies — 
1st. Firm 

belief; 
2d. Of what 

is true ; 

3d. On 
sufficient 
grounds. 



Examples. 



4'<J 



LOGIC. 



[book L 



FACTS AND TRUTHS. 



Knowledge is 



§ 24. Our knowledge is of two kinds : of facts 
of facts and anc { truths. A fact is any thins; that has been 

truths. J ° 

or is. That the sun rose yesterday, is a fact : 
that he gives light to-day, is a fact. That wa- 
ter is fluid and stone solid, are facts. We de- 
rive our knowledge of facts through the medium 
of the senses. 
Truth an Truth is an exact accordance with what has 

accordance 

with what been, is, or shall be. 1 here are two methods 

has been, is, r ... ' • 

or shaii be. of ascertaining truth : 
rwomethods i st g y comparing known tacts with each 

of ascertain- J ■ 

tag it other; and, 

2dly. By comparing known truths with each 
other. 

Hence, truths are inferences either from facts 
or other truths, made by a mental process called 
Reasoning. 



Facts and 
truths, the 

elements 

of our 
knowledge. 



§ 25. Seeing, then, that facts and truths are the 
elements of all our knowledge, and that knowl- 
edge itself is but their clear apprehension, their 
firm belief, and a distinct conception of their 
relations to each other, our main inquiry is, How 
are we to attain unto these facts and truths, 
which are the foundations of knowledge ? 

1st. Our knowledge of facts is derived through 



CHAP. II.] FACTS AND TRUTHS. 43 

the medium of our senses, by observation, exper- 
iment,* and experience. We see the tree, and Howwc 

arrive at a 

perceive that it is shaken by the wind, and note knowledge or 
the -fact that it is in motion. We decompose 
water and find its elements ; and hence, learn 
from experiment the fact, that it is not a simple 
substance. We experience the vicissitudes of 
heat and cold; and thus learn from experience 
that the temperature is not uniform. 

The ascertainment of facts, in any of the ways 
above indicated, does not point out any connec- This does not 
tion between them. It merely exhibits them to connection 
the mind as separate or isolated; that is, each ^^m* 
as standing for a determinate thing, whether 
simple or compound. The term facts, in the 
sense in which we shall use it, will designate 
facts of this class only. If the facts so ascer- 
tained have such connections with each other, when they 

. " . have a con- 

that additional facts can be inferred from them, nec tion that 

that inference is pointed out by the reasoning . 

* J o by the rea- 



is pointed out 

process, which is carried on, in all cases, by com- 



soning 
process. 

parison. 

2dly. A result obtained by comparing facts, we Truth, found 
have designated by the term Truth. Truths, yc ^^ mg 
therefore, are inferences from facts ; and every 

* Under this term we include all the methods of inves- 
tigation and processes of arriving at facts, except the pro- 
cess of reasoning. 



44 



LOG IC. 



[book I 



and 
is inferred 
from them. 



truth has reference to all the singular facts from 
which it is inferred. Truths, therefore, are re- 
sults deduced from facts, or from classes of facts. 
Such results, when obtained, appertain to all facts 
of the same class. Facts make a genus : truths, 
a species ; with the characteristic, that they be- 
come known to us by inference or reasoning. 



§ 26. How, then, are truths to be inferred 

There 



How 

truths are . c . . . 

inferred from * rom facts by the reasoning process? 

facts by the *— ^ ««™„ 

\ are two cases. 

reasoning 

1st. When the instances are so few and simple 



process. 
1st case. 



2d case. 



that the mind can contemplate all the fads on 
which the induction rests, and to which it refers, 
and can make the induction without the aid of 
other facts ; and, 

2dly. When the facts, being numerous, com- 
plicated, and remote, are brought to mind only 
by processes of investigation. 



Intuitive 

or 

Self-evident 

truths. 



Intuition 
defined. 



INTUITIVE TRUTH. 

§ 27. Truths which become known by con- 
sidering all the facts on which they depend, and 
which are inferred the moment the facts are 
apprehended, are the subjects of Intuition, and 
are called Intuitive or Self-evident Truths. The 
term Intuition is strictly applicable only to that 
mode of contemplation in which we look at 



CHAP. II.] INTUITIVE TRUTH. 45 



facts, or classes of facts, ani apprehend the 
relations of those facts at the same time, and 
by the same act by which we apprehend the 
facts themselves. Hence, intuitive or self-evi- How intuitive 

... truths are 

dent truths are those which are conceived in conceived in 
the mind immediately; that is, which are per- 
fectly conceived by a single process of induc- 
tion, the moment the facts on which they depend 
are apprehended, without the inteivention of 
other ideas. They are necessary consequences of 
conceptions respecting which they are asserted. Axioms of 
The axioms of Geometry afford the simplest and ^Ti^eT 
most unmistakable class of such truths. kind# 

"A whole is equal to the sum of all its parts," a whole 
is an intuitive or self-evident truth, inferred from 8um of d 
facts previously learned. For example; having; lhe P ar * s ' 

r J I V an intuitive 

learned from experience and through the senses truth - 
what a whole is, and, from experiment, the fact 
that it may be divided into parts, the mind per- 
ceives the relation between the whole and the 
sum of the parts, viz. that they are equal ; and 
then, by the reasoning process, infers that the How inferred 
same will be true of every other thing ; and 
hence, pronounces the general truth, that "a 
whole is equal to the sum of all its parts." Here 
all the facts from which the induction is drawn. A11 the f**** 

are presented 

are presented to the mind, and the induction to the mind. 
is made without the aid of other facts ; hence. 



46 LOGIC. [book I. 



aii the it is an intuitive or self-evident truth. All the 

axioms are . 

deduced in other axioms oi Ireometry are deduced irom 

the same . , , r . r , 

way , premises and by processes of inference, entirely 
similar. We would not call these experimental 
truths, for they are not alone the results of ex- 
periment or experience. Experience and exper- 
iment furnish the requisite information, but the 
reasoning power evolves the general truth. 

"When we say, the equals of equals are equal, 
we mentally make comparisons in equal spaces, 

These equal times, &c. ; so that these axioms, how- 
axioms are . 
general ever self-evident, are still general propositions: 

proposnons. so f ar r j| lc inductive kind, that, independently 
of experience, they would not present themsel 
to the mind. The only difference between th 
and axioms obtained from extensive inductioi 

Difference tliis : that, in raising the axioms of Geometry, 

them^Ki ^ ie i nstanccs °^ er themselves spontaneously, and 

othe , r without the trouble of search, and are few and 

propositions, 

which re- s ] m pi e ; [ n raising those of nature, they are in- 

(oirediligent i o j 

research, finitely numerous, complicated, and remote ; so 
that the most diligent research and the utmost 
acuteness are required to unravel their web. and 
place their meaning in evidence."* 



* Sir John Herschel's Discourse on the study of Natural 
Philosophy. 



CHAP. II.] LOGICAL TRUTHS. 47 

TRUTHS, OR LOGICAL TRUTHS. 

§ 28. Truths inferred from facts, by the process 
of generalization, when the instances do not offer Truths 

themselves spontaneously to the mind, but require from fact8> 

search and acuteness to discover and point out tn ^ in _ 

their connections, and all truths inferred from ferredfrom 

truths. 

truths, might be called Logical Truths. But as 
we have given the name of intuitive or self- 
evident truths to all inferences in which all the 
facts were contemplated, we shall designate all 
others by the simple term, Truths. 

It might appear of little consequence to dis- ff 
tinguish the processes of reasoning by which thedi stinc- 

& r . tion, being 

truths are inferred from facts, from those in which the basis of a 

classification. 

we deduce truths from other truths ; but this dif- 
ference in the premises, though seemingly slight, 
is nevertheless very important, and divides the 
subject of logic, as we shall presently see, into 
two distinct and very different branches. 

LOGIC. 

§ 29. Logic takes note of and decides upon Logic 
the sufficiency of the evidence by which truths s ^* nC y d 
are established. Our assent to the conclusion evidence, 
being grounded on the truth of the premises, we 
never could arrive at any knowledge by rea- 
soning, unless something were known antece- 
dently to all reasoning. It is the province of its province. 



48 LOGIC. [book I. 



Furnishes Logic to furnish the tests by which all truths 

tlic tests of 

truth. that are not intuitive may be inferred from the 
premises. It has nothing to do with ascertain- 
ing facts, nor with any proposition which claims 
to be believed on its own intrinsic evidence ; 
that is, without evidence, in the proper sense of 
Has nothing the word. It has nothing to do with the original 
intuitive pro- data, or ultimate premises of our knowledge ; 
positions, nor w j t h their number or nature, the mode in which 

with original 

data; they are obtained, or the tests by which they 
are distinguished. But, so far as our knowledge 
is founded on truths made such by evidence, 

but supplies 

all tests for that is, derived from farts or other truths pre- 
propositions. viously known, whether those truths be particu- 
lar truths, or general propositions, it is the prov- 
ince of Logic to supply the tests for ascertaining 
the validity of such evidence, and whether or 
not a belief founded on it would be well ground- 
ed. And since by far the greatest portion of 
The greatest our knowledge, whether of particular or general 

portion of our . . 

knowledge truths, is avowedly matter of inference, nearly 

conn, from the wh() j nQt Qn] of science but of human 
inference. J 

conduct, is amenable to the authority of logic. 



CHAP. II.] INDUCTION. 49 



INDUCTION. 

§ 30. That part of logic which infers truths 
from facts, is called Induction. Inductive rea- 
soning is the application of the reasoning pro- 



induction, 

to what 
reasoning 

cess to a given number of facts, for the purpose applicable. 
of determining if what has been ascertained re- 
specting one or more of the individuals is true 
of the whole class. Hence, Induction is not induction 
the mere sum of the facts, but a conclusion 
Irawn from them. 

The logic of Induction consists in classing Logic of 
the facts and stating the inference in such a 
manner, that the evidence of the inference shall 
be most manifest. 



§ 31. Induction, as above defined, is a process inducts 
of inference. It proceeds from the known to a^ae 
the unknown; and any operation involving no knowntoth9 

unknown. 

inference, any process in w T hich the conclusion 

is a mere fact, and not a truth, does not fall 

within the meaning of the term. The conclu- The conclu- 
sion broader 

sion must be broader than the premises. The than the 

r i i i premises, 

premises are tacts : the conclusion must be a 
truth. 

Induction, therefore, is a process of general- induction, 

t • i . n , , a process of 

ization. It is that operation of the mind by generaiiza- 
which we infer that what we know to be true tlon; 

4 



50 LOGIC. [book I. 



in which in a particular case or cases, will be true in all 

we conclude, , . , n , . r 

that what is cases which resemble the former in certain as- 
true under s jnmable respects. In other words, Induction is 

particular ° x 

circumstan- the process by which we conclude that what 

ces will be 

trueuniver- is true of certain individuals of a class is true 
of the whole class ; or that what is true at cer- 
tain times, will be true, under similar circum- 
stances, at all times. 



induction § 32- Induction always presupposes, not only 
presupposes fa^ ^ necessar y observations arc made with 

ucc urate and 

necessary the necessary accuracy, but also that the results 

observations. . , 

of these observations are, so far as practicable, 
connected together by general descriptions I ena- 
bling the mind to represent to itself as wholes, 
whatever phenomena are capable of being so 
represented. 

To suppose, however, that nothing more is 

More is required from the conception than that it should 

necessary serve to connect the observations, would be to 

than to 

connect the substitute hypothesis for theory, and imagina- 

observations: 

we must tion for proof. The connecting link must be 

infer from . 

ujeni. some character which really exists m the tacts 
themselves, and which would manifest itself 
therein, if the condition could be realized which 
our organs of sense require. 

For example ; Blakewell, a celebrated English 
cattle-breeder, observed, in a great number of 



CHAP. II.] INDUCTION. 51 

individual beasts, a tendency to fatten readily, Example of 
and in a great number of others the absence of the English 
this constitution : in every individual of the for- br c e a e "g r# 
mer description, he observed a certain peculiar 
make, though they differed widely in size, color, 
&c. Those of the latter description differed no 
less in various points, but agreed in being of a 
different make from the others. These facts were How he 
his data; from which, combining them with the the facts: 
genera] principle, that nature is steady and uni- wh *' ne 
form in her proceedings, he logically drew the 
conclusion that beasts of the specified make have 
universally a peculiar tendency to fattening. 

The principal difficulty in this case consisted in what the 

difficulty 

in making the observations, and so collating and consisted. 
combining them as to abstract from each of a 
multitude of cases, differing widely in many re- 
spects, the circumstances in which they all 
agreed. But neither the making of the observa- 
tions, nor their combination, nor the abstraction, 
nor the judgment employed in these processes, 
constituted the induction, though they were all 
preparatory to it. The Induction consisted in in what the 
the generalization ; that is, in inferring from all fja ^ eiAm 
the data, that certain circumstances would be 
found in the whole class. 

The mind of Newton was led to the universal 
law, that all todies attract each other by force* 



52 



LOGIC. 



[book I. 



Newton's 

inference of 

the law of 

universal 

gravitation. 



How he 

observed 

facts and 

their 

connections. 



The use 
which he 
made of 
exact 
science. 



What was 
the result. 



varying directly as their masses, and inversely 
as the squares of their distances, by Induction. 
He saw an apple falling from the tree : a mere 
fact ; and asked himself the cause ; that is, if any 
inference could be drawn from that fact, which 
should point out an invariable antecedent condi- 
tion. This led him to note other facts, to prose- 
cute experiments, to observe the heavenly bodies, 
until from many facts, and their connections 
with each other, he arrived at the conclusion, 
that the motions of the heavenly bodies were gov- 
erned by general laws, applicable to all matter , 
that the stone whirled in the sling and the earth 
rolling forward through space, are governed in 
their motions by one and the same law. lie 
then brought the exact sciences to his aid. and 
demonstrated that this law accounted for all the 
phenomena, and harmonized the results of all ob- 
servations. Thus, it was ascertained that the 
laws which regulate the motions of the heav- 
enly bodies, as they circle the heavens, also 
guide the feather, as it is wafted along on the 
passing breeze. 



rhe ways of §33. We have already indicated the ways in 

facts are° which the facts are ascertained from which the 

known: i n f erences are drawn. But when an inference 

can be drawn ; how many facts must enter into 



CHAP. II.] 



INDUCTION. 



53 



the premises ; what their exact nature must be ; 
and what their relations to each other, and to 
the inferences which flow from them ; are ques- 
tions which do not admit of definite answers. 
Although no general law has yet been discov- 
ered connecting all facts with truths, yet all the 
uniformities which exist in the succession of phe- 
nomena, and most of those which prevail in their 
coexistence, are either themselves laws of cau- 
sation or consequences resulting and corollaries 
capable of being deduced from, such laws. It 
being the main business of Induction to deter- 
mine the effects of every cause, and the causes 
of all effects, if we had for all such processes 
general and certain laws, we could determine, 
in all cases, what causes are correctly assigned 
to what effects, and what effects to what causes, 
and we should thus be virtually acquainted with 
the whole course of nature. So far, then, as we 
can trace, with certainty, the connection be- 
tween cause and effect, or between effects and 
their causes, to that extent Induction is a sci- 
ence. When this cannot be done, the conclu- 
sions must be, to some extent, conjectural. 



but we 

do not know 

certainly, 

in all cases, 

when we can 

draw on 

inference. 



No 

general law. 



Business 

of 
Induction. 



What is 
necessary. 



How far a 
science. 



54 



LOGIC. 



[book I. 



CHAPTER III. 



DEDUCTION NATURE OF THE SYLLOGISM ITS USES AND APPLICATIONS. 



Inductive 
processes of 
reasoning. 



Deductive 
processes. 



Deduction 
defined. 



Deductive 
formula. 



DEDUCTION. 

§ 3 4. We have seen that all processes of 
Reasoning, in which the premises are particular 
facts, and the conclusions general truths, are 
called Inductions. All processes of Reasoning, 
in which the premises are general truths and the 
conclusions particular truths, are called Deduc- 
tions. Hence, a deduction is the process of 
reasoning by which a particular truth is inferred 
from other truths which are known or admitted. 
The formula for all deductions is found in the 
Syllogism, the parts, nature, and uses of which 
we shall now proceed to explain. 



TROrOSITIONS. 



Proposition, § 35 - ^ proposition is a jiidgmeM expressed 

judgment in ^ n wor( j s Hence, a proposition is defined lop-i- 

words : 

cally, "A sentence indicative:" affirming or 



* Section 30. 



CHAP. III.] PROPOSITIONS. 55 



denying; therefore, it must not be ambiguous, must not be 

c , i'ii 1 • ambiguous; 

tor that which has more than one meaning is norimper . 
in realitv several propositions: nor imperfect, fect; nurim " 

■ l l ■*■ J grammatical. 

nor an grammatical, for such expressions have 
no meaning at all. 



§ 36. Whatever can be an object of belief, 
or even of disbelief, must, when put into words, a proposition 
assume the form of a proposition. All truth and exp 
all error lie in propositions. What we call a 
truth, is simply a true proposition; and errors its nature,^ 

extent. 

are false propositions. To know the import of 
all propositions, would be to know all questions 
which can be raised, and all matters which are Embraces an 

truth and all 

susceptible of being either believed or disbe- error, 
lieved. Since, then, the objects of all belief and 
all inquiry express themselves in propositions, a 
sufficient scrutiny of propositions and their va- An examina- 
rieties will apprize us of what questions mankind propositions 
have actually asked themselves, and what, in the em ^ ce9al \ 

7 ■ questions and 

nature of answers to those questions, thev have a 11 ^ ^- 

edge. 

actually thought thev had grounds to believe. 



§ 37. The first glance at a proposition shows a proposition 
that it is formed by putting together two names. 
Thus, in the proposition, "Gold is yellow, " the 
property yellow is affirmed of the substance gold. 
In the proposition, •'• Franklin was not born in 



putting t-w o 

names 

together. 



5tt 



LOGIC. 



[book 



England," the fact expressed by the words born 
in England is denied of the man Franklin. 



A 
proposition 
has 111 roe 

parts: 

Bulged) 

P rmH r rtfr 

ami 
Copula. 



Subject 
defined. 



Triplicate. 



( »pwla 
must bfl 

is or 18 NOT. 

All verbs 
resolvable 

koto ' k to be." 



§ 38. Every proposition consists of three 
parts : the Subject, the Predicate, and the Co- 
pula. The subject is the name denoting the 
person or thing of which something is affirmed 
or denied : the predicate is that which is affirm- 
ed or denied of the subject ; and these two are 
called the terms (or extremes), because, logically, 
the subject is placed first, and the predicate last. 
The copula, in the middle, indicates the act of 
judgment, and is the sign denoting that there is 
an affirmation or denial. Thus, in the proposi- 
tion, " The earth is round ;" the subject is the 
words " the earth," being that of which some- 
thing is affirmed: the predicate, is the word round, 
which denotes the quality affirmed, or (as the 
phrase is) predicated : the word is, which serves 
as a connecting mark between the subject and 
the predicate, to show that one of them is af- 
firmed of the other, is called the Copula. The 
copula must be either is, or is not, the substan- 
tive verb being the only verb recognised by 
Logic. All other verbs are resolvable, by means 
of the verb M to be," and a participle or adjective. 
For example : 

• ; The Romans conquered :" 






CHAT. III.] 



B1 L LOG ISM. 



r>7 



the word "conquered?' is both copula and predi- examples 

, . , . . . ,, of the 

cate, being equivalent to " were victorious. comua. 
Hence, we might write, 



•* The Romans were victorious,"' 

in which were is the copula, and victorious the 
predicate. 

§ l 3\). V proposition being a portion of dis- Apinp«Wo« 

is either 

course, in which something is affirmed or denied affirmative 

oi something, all propositions may be divided 
into affirmative arid negative. An affirmative 
proposition is that in which the predicate is af- 
firmed of the subject; as, "Caesar is dead." A 
_ itive proposition is that in which the predicate 
is denied of the subject ; as, " Caesar is not dead.'' 
The copula, in this last species of proposition. latheM, 

of the words "is \ht ; " which is the 
sign of negation ; " is" being the sign of affirm- 
ation. 



llM '-"['ill;! IS. 



SYLLOGISM. 



§ 40. A syllogism is a form of stating the con- \ qUaglm 
nection which may exist, for the purpose of three prape 
loning, between three propositions. Hence, 



to a legitimate syllogism, it is essential that 

there should be three, and only three, proposi* «d«itted: 



I 



58 logic. [book I. 

and the third tions. Of these, two are admitted to be true, 

fromijlcm. anc * are ca ^ e( l tne P rem ^ ses •* the third is proved 
from these two, and is called the conclusion. 
For example : 



Example. 



11 All tyrants are detestable : 
Caesar was a tyrant ; 
Therefore, Caesar was detestable." 



Major Term 



Now, if the first two propositions be admitted 

the third, or conclusion, necessarily follows from 

them, and it is proved that Cjssab was detestable. 

Of the two terms of the conclusion, the Predi- 

deflned. cate (detestable) is called the major term, and 

the Subject (Caesar) the minor term ; and these 

two terms, together with the term "tyrant/' 

make up the three propositions of the syllogism, 

Minor Term. — each term being used twice. Hence, every 

syllogism has three, and only three, different 

terms. 

Major r fhe premiss, into which the Predicate of the 

Premiss 

defined, conclusion enters, is called the major premiss; 

Minor the other is called the minor premiss, and con- 

Premiss. ta j ns tne Subject of the conclusion ; and the 

other term, common to the two premises, and 

with which both the terms of the conclusion were 

separately compared, before they were compared 

MiddieTerm. w jth each other, is called the middle term. In 

the syllogism above, "detestable" (in the cor 



CHAP. III.] SYLLOGISM. 59 



elusion) is the major term, and " Caesar" the mi- Example, 

. pointing out 

nor term : hence, Major 

premiss, 
" All tyrants are detestable," Minor 

. . premiss, and 

is the major premiss, and Middle Term. 

" Caesar was a tyrant," 
the minor premiss, and " tyrant" the middle term. 

§ 41. The syllogism, therefore, is a mere for- 

* J & > » Syllogism, 

mula for ascertaining what may, or what may a mere 

formula. 

not, be predicated of a subject. It accomplishes 

this end by means of two propositions, viz. by 

comparing the given predicate of the first (a How applied. 

Major Premiss), and the given subject of the 

second (a Minor Premiss), respectively with one 

and the same third term (called the middle term), 

and thus — under certain conditions, or laws of 

the syllogism — to be hereafter stated — eliciting 

the truth (conclusion) that the given predicate 

must be predicated of that subject. It will be use of the 

seen that the Major Premiss always declares, premiss. 

in a general way, such a relation between the 

Major Term and the Middle Term ; and the Mi- of the Minor 

nor Premiss declares, in a more particular way, 

such a relation between the Minor Term and 

the Middle Term, as that, in the Conclusion, of the 

i -t/r- m i i i -»r • Middle Term, 

the Minor lerm must be put under trie Major 
Term ; or in other words, that the Major Term 
must be predicated of the Minor Term. 



60 logic. [book 1. 



ANALYTICAL OUTLINE OF DEDUCTION. 

Reasoning § 42. In every instance in which we reason, 
in the strict sense of the word, that is, make use 
of arguments, whether for the sake of refuting 
an adversary, or of conveying instruction, or of 
satisfying our own minds on any point, whatever 
may be the subject we are engaged on. a certain 
process takes place in the mind, which is one 
The process, and the same in all cases (provided it be COT- 

the same. I'^ctly conducted), whether we use the inductive 
process or the deductive formulas. 

Of course it cannot lie Supposed thai every 

Ereryooe QQe is even conscious of this process in his own 

not conscious i i i • i • . i 

ofthe mind: much less, )- competent to explain the 
process, principles on which it proceeds. This indeed is, 
The same tor and cannot but he. the case with every other 
process. Process respecting which any system has been 
formed ; the practice not only may exist inde- 
pendently of the theory, but must have preceded 
the theory. There must have been Language 
Eiementsand before a system of Grammar could be devised ; 
knowledge of an( j mus i ca ] compositions, previous to the sci- 

elemcnts, x 1 

must precede ence of Music. This, by the way, serves to ex- 

generaliza- 
tion and pose the futility of the popular objection against 

of principles. Logic ; viz. that men may reason very well who 
know nothing of it. The parallel instances ad- 
duced show that such an objection may be urged 



CHAP. 111.] ANALYTICAL OUTLINE. 01 

in many other cases, where its absurdity would Logic 
be obvious ; and that there is no ground for de- 
ciding thence, either that the system has no ten- 
dency to improve practice, or that even if it had 
not, it might not still be a dignified and inter- 
esting pursuit. 

§ 43. One of the chief impediments to the sameoeas of 

the reasoning 

attainment of a just view of the nature and ob- process 
ject of Logic, is the not fully understanding, or kept in mind, 
not sufficiently keeping in mind the sameness 
of the reasoning process in all cases. If, as the 
ordinary mode of speaking would seem to indi- 
cate, mathematical reasoning, and theological, ah kinds of 
and metaphysical, and political, &c, were essen- "^^L"" 
tially different from each other, that is, different r rinci P le 
kinds of reasoning, it would follow, that suppo- 
sing there could be at all any such science as 
we have described Logic, there must be so many 
different species or at least different branches 
of Logic. And such is perhaps the most pre- 
vailing notion. Nor is this much to be won- Beaaonof 

... mi the prevail' 

tiered at ; since it is evident to all, that some ing errori 
men converse and w T rite, in an argumentative 
way, very justly on one subject, and very erro- 
neously on another, in which again others excel, 
who fail in the former. 

This error may be at once illustrated and re- 



62 logic. [book I. 

The reason of moved, by considering the parallel instance of 
illustrated Arithmetic ; in which every one is aware that 
by example, ^ p rocess f a calculation is not affected by 

which shows L J 

that the rear the nature of the objects whose numbers are 

soiling 

process is before us; but that, for example, the multipli- 
lame. " cation of a number is the very same operation, 
whether it be a number of men, of miles, or of 
pounds; though, nevertheless, persons may per- 
haps be found who are accurate in the results 
of their calculations relative to natural philoso- 
phy, and incorrect in those of political econo- 
my, from their different degrees of skill in the 
subjects of these two sciences; not Rifely lo- 
calise there are different arts of arithmetic ap 
plicable to each of these respectively. 

§ 44. Others again, who are aware that the 
some view wnpfo system of Logic may be applied to all 

Logic as a 

peculiar subjects whatever, arc yet disposed to view it 

method of . . , . r . . 

reasoning: as a peculiar method oi reasoning, and not, as 
it is, a method of unfolding and analyzing our 
reasoning : whence many have been led to talk 
of comparing Syllogistic reasoning with Moral 
reasoning ; taking it for granted that it is pos- 
sible to reason correctly without reasoning logi- 

it is the only cally | which is, in fact, as great a blunder as if 

method of . ■. c 

any one were to mistake grammar lor a pecu- 



reasonmg 



correctly, j^ j an g Ua g Ci and to sup|)ose it possible to speak 



CHAP. III.] ANALYTICAL OUTLINE 63 

correctly without speaking grammatically. They 
have, in short, considered Logic as an art of rea- 
soning ; whereas (so far as it is an art) it is the 
art of reasoning; the logician's object being, not it lays down 

rules, not 

to lay down principles by which one may reason, which may, 
but by which all must reason, even though they must be 
are not distinctly aw T are of them : — to lay down 
rules, not which may be followed with advan- 
tage, but which cannot possibly be departed 

from in sound reasoning. These misapprehen- Misappre- 
hensions and 
sions and objections being such as lie on the objections 

very threshold of the subject, it would have been 
hardly possible, without noticing them, to con- 
vey any just notion of the nature and design of 
the logical system. 

§45. Supposing it then to have been per- operation of 
ceived that the operation of reasoning is in all should be 
cases the same, the analysis of that operation anal >' zed: 
could not fail to strike the mind as an interesting 
matter of inquiry. And moreover, since (appa- 
rent) arguments, which are unsound and incon- 
clusive, are so often employed, either from error Because such 
or design; and since even those who are not ^1!!!^ 

° necessary to 

misled by these fallacies, are so often at a loss furnisb the 
to detect and expose them in a manner satis- 
factory to others, or even to themselves ; it could 
not but appear desirable to lay down some gen- 



64 logic. [book I 



/uicsforthc eral rules of reasoning, applicable to all OS 

!T«? by which a person misjht be enabled the more 

error and the J i o 

discovery of rC adily and clearly to state the grounds of his 

truth. 

own conviction, or of his objection to the argu- 
ments of an opponent; instead of arguing at 
random, without ;my fixed and acknowledged 

principles to guide Ids procedure. Such rules 

such rules wou | ( | ] )r analogous to those of Arithmetic, which 
■re inuogoui 

to the rules oi obviate the tediousnesa and uncertainty of cal- 

Arithmetie. 

dilations in the head: wherein, after much labor, 



different persons might arrive at different results, 

without any of them being able distinctly to 

point out the error of the rest. A system ot 

such rules, it is nl>\i<>us. must, instead ofdeserv- 

They bring ing to be called the art of wrangling. I»e more 

the parlies, in . . ■ i i r • I 

arKlim ,. Mti to justly characterized as the "art ot cutting short 
an issue. W rangling," 1 >y bringing the parties to issue at 

once, it' not to agreement; and thus Baying a 

waste of ingenuity. 



Every con- § 1(>. In pursuing the supposed investigation, 

elusion is 

deduredtrom it will be touud tliat. in all deductive prOOtt 

wopropoBi- ever y conclusion is deduced, in realitv. from two 

tions, called 

Premises, other propositions (thence called Premises) ; for 

itoneprem- t lion^li one o( these may be, and conimonlv is, 

iss is sup- ° J J 

pressed, it is suppressed, it must nevertheless be understood 
nevertheless 

understo.nl, as admitted; as may easily be made evident by 

supposing the denial of the suppressed premiss, 



CHAP. HI.] ANALYTICAL OUTLINE. 65 

which will at once invalidate the argument. For 
example ; in the following syllogism : 

•• Whatever exhibits marks of design had an intelligent author: 
The world exhibits marks of design ; 
Therefore, the world had an intelligent author :" 

if any one from perceiving that " the world ex- 
hibits marks of design/' infers that "it must have 3^^ 
had an intelligent author/' though he may not be n ^^° 
aware in his own mind of the existence of any menu though 

one m3y not 

other premiss, he will readily understand, if it be be aware 

led that "whatever exhibits marks of design 
must have had an intelligent author,'' that the 
affirmative o( that proposition is necessary to 
the validity of the argument. 



547. When one of the premises is suppressed Enthymeme-. 

a syllogism 

(which for brevity's sake it usually is), the argu- with oue 
ment is called an Enthymeme. For example : sup^rcawd. 

u The world exhibits mark* of deaig 

Therefore the world had an intelligent antfa 

is an Enthymeme. And it may be worth while 

to remark, that, when the argument is in this objections 

. , made to the 

state, the objections of an opponent are (or rather MJJ „ ffcaor 
appear to be) of two kinds, viz. either objections ^^e^Z, 
to the assertion itself, or objections to its force meut * 
an argument. For example: in the above Example, 
instance, an atheist m'av be conceived either de- 



C6 logic. [book I. 

Both prem- n y m § t ^ lat ^he world does exhibit marks of de- 
ises must be s ig n or denying that it follows from thence that 

true, if the ° J ° J 

argument is it had an intelligent author. Now it is impor- 

sound : 

tant to keep in mind that the only difference in 

the two cases is, that in the one the expressed 

premiss is denied, in the other the suppressed,; 

and when for the force as an argument of either premiss 

tbo cMftriw depends on the other premiss : if both be admit- 

8ion follows. te j the conclusion legitimately connected with 

them cannot be denied. 

§ 48. It is evidently immaterial to the argu- 
ment whether the conclusion be placed first or 
PremiM Inst; but it. may be proper to remark, thai a 
thecoDcio- premiss placed (tfter its conclusion is called the 

lion is called j{ eason f j t an( i j s introduced by one of those 

the Reason. J 

conjunctions which are called causal, viz. k * since/' 

" because," &c, which may indeed be employed 

to designate a premiss, whether it come first or 

niative ta st - Tl ie illative conjunctions "therefore," &c, 

conjunction. des j gnate the conclusion. 

It is a circumstance which often occasions 
causes of error all( j perplexitv, that both these classes of 

error and l A J 

perplexity, conjunctions have also another signification, be- 
ing employed to denote, respectively, Cause and 
Effect, as w r ell as Premiss and Conclusion. For 

Different M 

significations example ; if I say, " this ground is rich, because 

of the . . . 

^junctions, the trees on it are nourishing; or. " the trees are 






CHAP. III.] ANALYTICAL OUTIINE. 67 

flourishing, and therefore the soil must be rich ;" Examples 
I employ these conjunctions to denote the con- CODjunctiong 
nection of Premiss and Conclusion; for it is ^ reus ^ d 

logically. 

plain that the luxuriance of the trees is not the 
cause of the soil's fertility, but only the cause 
of my knowing it. If again I say, "the trees 
flourish, because the ground is rich ;" or " the 
ground is rich, and therefore the trees flourish/ Examples 

T . , . -. where they 

1 am using the very same conjunctions to denote denote 



cause 
and effect. 



the connection of cause and effect; for in this 
case, the luxuriance of the trees being evident 
to the eye, would hardly need to be proved, but 
might need to be accounted for. There are, Many 

i i • i ,i . in which the 

however, many cases, in which the cause is em- cause and 
ployed to prove the existence of its effect ; espe- the reasou 

1 J x l are the same, 

cially in arguments relating to future events: as, 
for example, when from favorable weather any 
one argues that the crops are likely to be abun- 
dant, the cause and the reason, in that case, co- 
incide ; and this contributes to their being so 
often confounded together in other cases. 



•*Br 



§ 49. In an argument, such as the example i u every 

orrect argu 
ment, to 



above given, it is, as has been said, impossible C( 



for any one, who admits both premises, to avoid admit the 

premiss is tc 

admitting the conclusion. But there will be fre- admit the 

. conclusion. 

quently an apparent connection of premises with 
a conclusion which does not in realitv follow 



68 logic. [book i. 



Apparent from them, though to the inattentive or unskilful 
premises and the argument may appear to be valid ; and there 
conclusion are man y ther cases in which a doubt may exist 

must not be J J 

relied on. whether the argument be valid or not ; that is, 
whether it be possible or not to admit the prem- 
ises and yet deny the conclusion. 



General rules § 50. It is of the highest importance, there- 
for argil men- 



tation 



fore, to lay down some regular form to which 
necessary. ever y va lid argument may be reduced, and to 
devise a rule which shall show the validity of 
every argument in that form, and consequently 
the unsoundness of any apparent argument which 
cannot be reduced to it. For example J if such 
an argument as this be proposed: 



Example of '* Every rational agent is accountable : 

an imperfect Brutes arc not rational agents ; 

Therefore they arc not accountable ;" 



argument. 



or again : 

«M Example. " All wise legislators suit their laws to the genius of their 
nation ; 
Solon did this; therefore he was a wise legislator :" 

Difficulty of there are some, perhaps, who would not per- 
letectmgthe ce j ve any f a u aC y j n suc h arguments, especially 

enor. * • ° * ■ 

if enveloped in a cloud of words ; and still more, 
w T hen the conclusion is true, or (which comes to 
the same point) if they are disposed to believe 
it ; and others might perceive indeed, but might 



CHAP. III.] ANALYTICAL OUTLINE. C9 



be at a loss to explain, the fallacy. Now these Towhat 

, 1 these appa- 

(apparent) arguments exactly correspond, re- reut 
spectively, with the following, the absurdity of ^^ 
the conclusions from which is manifest : 

" Every horse is an animal : A similar 

Sheep are not horses ; 
Therefore, they are not animals." 

And: 

" All vegetables grow ; 2d similar 

. . t example. 

An animal grows ; 

Therefore, it is a vegetable." 
These last examples, I have said, correspond These last 

\ • i i r* correspond 

exactly (considered as arguments) with the lor- wilh th0 
mer ; the question respecting the validity of an former - 
argument being, not whether the conclusion be 
true, but whether it follows from the premises 
adduced. This mode of exposing a fallacy, by This mode of 
brino-incr forward a similar one whose conclusion , * xpo6ing 

o O fallacy some- 

is obviously absurd, is often, and very ad van- times 

resorted to. 

tageously, resorted to in addressing those who 
are ignorant of Logical rules ; but to lay down 
such rules, and employ them as a test, is evi- To lay down 
dently a safer and more compendious, as well best way 
as a more philosophical mode of proceeding. To 
attain these, it would plainly be necessary to 
analyze some clear and valid arguments, and to 
observe in what their conclusiveness consists. 



70 LOGIC. [liOOK I 

§ 51. Let us suppose, then, such an examin- 
ation to be made of the syllogism above men- 
tioned : 

Example of " Whatever exhibits marks of design had an intelligent author; 
a perfect The worM exhib j ts markg of des ; n . 

syllogism. 

Therefore, the world had an intelligent author.*' 

what is Iii the first of these premises we find it as- 

the first sumed universally of the class of "things which 
premiss, exhibit marks of design/ 2 that they had an intel- 
in the second ligent author ; and in the other premiss, "the 
world" is referred to that class as comprehended 
whatwe m j t . now jf- j s evident that what said of 

may infer. 

the whole of a class, may be said of any thing 
comprehended in that class ; sq that we arc thus 
authorized to say of the world, that "it had an 
intelligent author."' 
Syllogism Again, if we examine a syllogism with a 

with a . . 

iK-ative negative conclusion, as, lor example, 

conclusion. 

" Nothing which exhibits marks of design could have been 

produced by chance ; 
The world exhibits, &c. ; 
Therefore, the world could not have been produced by 

chance," 

The process the process of reasoning will be found to be the 

of reasoning 

the same, same; since it is evident that whatever is denied 
universally of any class may be denied of any 
thing that is comprehended in that class. 



CHAP. IF.] ANALYTICAL OUTLINE. 71 



§ 52. On further examination, it will be found ah valid 
that all valid arguments whatever, which are re d UC ibieto 
based on admitted premises, maybe easily re- l e \^ SlC 
duced to such a form as that of the foregoing 
syllogisms; and that consequently the principle 
on which they are constructed is that of the for- 
mula of the syllogism. So elliptical, indeed, is the 
ordinary mode of expression, even of those who 0rdinar y 

J r mode of 

are considered as prolix WTiters, that is, so much expressing 

arguments 

is implied and left to be understood in the course elliptical. 
of argument, in comparison of what is actually 
stated (most men being impatient even, to excess, 
of any appearance of unnecessary and tedious 
formality of statement), that a single sentence 
will often be found, though perhaps considered 
as a single argument, to contain, compressed 
into a short compass, a chain of several distinct 
arguments. But if each of these be fully devel- But when 

& J fully devel 

oped, and the whole of what the author intended oped, they 

it i i • -n i r i i ma >' a11 be 

to imply be stated expressly, it will be found that re( iuced into 
all the steps, even of the longest and most com- tbeabove 

r ' w form. 

plex train of reasoning, may be reduced into the 
above form. 



§ 53. It is a mistake to imagine that Aristotle 
and other logicians meant to propose that this Aristotle did 

° x L not mean 

prolix form of unfolding arguments should uni- that every 

argument 

versally supersede, in argumentative discourses, should be 



72 logic. [book I 



thrown into the common forms of expression ; and that " to 

the form of a . ,, .. 

syllogism, reason logically, means, to state all arguments 
at full length in the syllogistic form ; and Aris- 
totle has even been charged with inconsistency 
for not doing so. It has been said that he " ar- 
gues like a rational creature, and never attempts 
That form is to bring his own system into practice." As well 

tftJtfk might a chemist be charged with inconsistency 
for making use of any of the compound sub- 
stances that are commonly employed, without 
previously analyzing and resolving them into 

Analogy to their simple elements; as well might it be ini- 

the chemist. 

agined that, to speak grammatically, means, to 
parse every sentence we utter. The chemist 

(to pursue the illustration) keeps by him his ti 
and his method of analysis, to be employed when 
Tbe analogy any substance is offered to his notice, the COm- 

continued. . . r . . . . . 

position of which lias not been ascertained, or 
in which adulteration is suspected. Now a fal- 

Towhata lacy may aptly be compared to some adulterated 

fallacy may - . . r 

becomparcd. compound ; "it consists ot an ingenious mixture 
of tiuth and falsehood, so entangled, so intimate- 
ly blended, that the falsehood is (in the chemical 
phrase) held in solution : one drop of sound logic 

yow detect- * s ^hat test which immediately disunites them, 
ed - makes the foreign substance visible, and precipi- 
tates it to the bottom." 



whaf. iii.] analytical outline. 73 

aristotle's dictum. 

J 54. But to resume the investigation of the Form of 
pi uiciples of reasoning : the maxim resulting from a e r I^^ t# 
the examination of a syllogism in the foregoing 
form, and of the application of which, every valid 
deduction is in reality an instance, is this : 

" That whatever is predicated (that is, affirmed Aristotle's 

dictum, 

or denied) universally, of any class of things, 
m\j be predicated, in like manner (viz. affirmed 
oi denied), of any thing comprehended in that 
c'"\ss." 

This is the principle commonly called the die- what the 
turn de omni et nullo, for the indication of pru ^J po 

7 is called. 

which we are indebted to Aristotle, and which 

is the keystone of his whole logical system. It 

is remarkable that some, otherwise judicious Whatwritera 

writers, should have been so carried away bv havesaid ° f 

this princi- 

their zeal against that philosopher, as to speak pie; and 

why. 

with scorn and ridicule of this principle, on 
account of its obviousness and simplicity : . ,. ... 

r J '■ Simplicity a 

though they would probably perceive at once testof 

science. 

in any other case, that it is the greatest tri- 
umph of philosophy to refer many, and seem- 
ingly very various phenomena to one, or a very 
few, simple principles ; and that the more simple 
and evident such a principle is, provided it be 
truly applicable to all the cases in question, the 



74 LOGIC. [book I. 



No solid ob- greater is its value and scientific beauty. If, 

jection to the . -, -, . . , , . , , 

principle indeed, an y principle be regarded as not thus ap- 
ever urged. p]j ca bi ej ij la i ^ an objection to it of a different 
kind. Such an objection against Aristotle's dic- 
tum, no one has ever attempted to establish by 
been taken an y kind of proof ; but it has often been taken 
forgranted. j or g ran t e d ; it being (as has been stated) very 
syllogism commonly supposed, without examination, that 

not a distinct . »••»•»* 

kind of ar- the syllogism is a distinct kind of argument and 

gU Tform bUt ^ at ^ e ru ' es °f & accordingly do not apply, nor 
applicable to were intended to applv, to all reasoning what 

all cases. 

ever, where the premises are granted or known. 



objection: § 55. One objection against the dictum of Aris- 
tbatthesyi- tQt j e ft m ^ wcx fa w i£fe to noi [ ce briefly, for 

logism was J J 

intended to the sake of setting in a clearer light the real 

make a dera- 

rustration character and object of that principle. The ap- 
plication of the principle being, as has been 
seen, to a regular and conclusive syllogism, it 
has been urc;ed that the dictum was intended 
to prove and make evident the conclusiveness 
of such a syllogism; and that it is unphilo- 
sophical to attempt giving a demonstration of 
a demonstration. And certainly the charge 

to increase would be just, if we could imagine the logi- 
cian's object to be, to increase the certainty 
of a conclusion, which we are supposed to have 
already arrived at by the clearest possible mode 



the certainty 

of a 
conclusion. 






CHAP. III.] ANALYTICAL OUTLINE. 75 



of proof. But it is very strange that such an This view is 
idea should ever have occurred to one who had erroneous, 
even the slightest tincture of natural philosophy ; 
for it might as well be imagined that a natural illustration. 
philosopher's or a chemist's design is to strength- 
en the testimony of our senses by a priori rea- 
soning, and to convince us that a stone when 
thrown will fall to the ground, and that gunpow- 
der will explode w T hen fired ; because they show 
according to their principles those phenomena 
must take place as they do. But it would be 
reckoned a mark of the grossest ignorance and 

. , . t , i • i • • The ob J ect is 

stupidity not to be aware that their object is nol to prove, 
not to prove the existence of an individual butt ° ac ~ 

^ count for 

phenomenon, which our eyes have witnessed, 
but (as the phrase is) to account for it ; that is, 
to show according to w T hat principle it takes 
place ; to refer, in short, the individual case to 
a general law of nature. The object of Aris- Theob J ectof 

° J the Dictum 

totle's dictum is precisely analogous: he had, to point out 

the general 

doubtless, no thought of adding to the force of process to 
any individual syllogism ; his design was to point ^^ c ^ 
out the general principle on which that process forms - 
is conducted which takes place in each syllo- 
gism. And as the Laws of nature (as they are Laws of 
called) are in reality merely generalized facts, of erased facts, 
which all the phenomena coming under them are 
particular instances ; so, the proof drawn from 



16 LOGIC. [BOOK I 



The Dictum Aristotle's dictum is not a distinct demonstration 
form of ail brought to confirm another demonstration, but is 
demonsira- mere ]y a generalized and abstract statement of 
all demonstration whatever; and is, therefore, in 
fact, the very demonstration which, under proper 
suppositions, accommodates itself to the various 
subject-matters, and which is actually employed 
in each particular case. 

now to trace §56. In order to trace more distinctly the 

the abstract- . » 

ingami dillerent steps of the abstracting process, by 
" ,:i> "" m " which any particular argument mav be brought 

proa—. J r © J 

into the most general form, we may first take a 

syllogism, that is, an argument stated accurately 

Annrgument and at full length, such as the example formerly 

stated at full 

length. given : 

" Whatever exhibits marks of design had an intelligent author; 
The world exhibits marks of design ; 
Therefore, the world had an intelligent author:" 

Propositions an d tnen somewhat generalize the expression, by 
expressed by su "b s tituthiff (as in Algebra) arbitrary unmean- 

abstract ° v o / J 

terms. j np - svmbols for the significant terms that were 
originally used. The syllogism will then stand 
thus : 

" Every B is A ; C is B ; therefore C is A." 

The reason- The reasoning, when thus stated, is no less evi- 
vaiid, dently valid, whatever terms A. B. and C respect- 



CHAP. III.] ANALYTICAL OUTLINE. 77 



ively may be supposed to stand for ; such terms and 
may indeed be inserted as to make all or some general, 
of the assertions false ; but it will still be no less 
impossible for any one who admits the truth of 
the premises, in an argument thus constructed, 
to deny the conclusion ; and this it is that con- 
stitutes the conclusiveness of an argument. 

Viewing, then, the syllogism thus expressed, SyflogfemBo 

viewed, 

it appears clearly that " A stands for any thing affirms gen- 
whatever that is affirmed of a certain entire class" between the 
(viz. of every B), " which class comprehends or terms * 
contains in it something else" viz. C (of which B 
is, in the second premiss, affirmed) ; and that, 
consequently, the first term (A) is, in the conclu- 
sion, predicated of the third (C). 



§ 57. Now, to assert the validity of this pro- Another form 

, r . , , . of stating the 

cess now before us, is to state the very dictum dictum, 
we are treating of, with hardly even a verbal 
alteration, viz. : 

1. Any thing whatever, predicated of a whole The three 

, things 

°i ass ' implied. 

2. Under which class something else is con- 
tained ; 

3. May be predicated of that which is so con- 
tained. Thesethree 

members 

The three members into which the maxim is correspond to 

the three 

here distributed, correspond to the three propo- proposition 



78 logic. [book I 

sitions of the syllogism to which they are in- 

ml O mt 

tended respectively to apply. 
Advantage of The advantage of substituting for the terms, 

substituting • i 11 • i •, 

arbitrary nl a regular syllogism, arbitrary, unmeaning sym- 
symboisfor b ] s> suc h as letters of the alphabet, is much the 

the terms. A 

same as in geometry : the reasoning itself is then 
considered, by itself, clearly, and without any 
risk of our being misled by the truth or falsi ty 
of the conclusion ; which is, in fact, accidental 
and variable; the essential point being, as far 
connection, ^ argument is concerned, the connection be- 

the essentia] 

point ofthe (ween the premises and the conclusion. We are 

argument. 

thus enabled to embrace the genera] principle of 
deductive reasoning, and to perceive its appli* 
bility to an indefinite number of individual cat 
That Aristotle, therefore, should have been ac- 
Aristotie cusec i n [* making use of these symbols for the 

right in using J 

tttesesym- purpose of darkening his demonstrations, and 
that too by persons not unacquainted with geom- 
etry and algebra, is truly astonishing. 



bols. 



Btract terms 
are used. 



syllogism § 58. It belongs, then, exclusively to a syllo- 

cqually true 

whenab- gism, properly so called (that is, a valid argu- 
ment, so stated that its conclusiveness is evident 
from the mere form of the expression), that if 
letters, or any other unmeaning symbols, be sub- 
stituted for the several terms, the validity of the 
argument shall still be evident. Whenever this 



CHAP. III.] ANALYTICAL OUTLINE. 79 

is not the case, the supposed argument is either when not so, 
unsound and sophistical, or else may be reduced argument 
(without any alteration of its meaning) into the u unsound 
syllogistic form ; in which form, the test just 
mentioned may be applied to it. 

§ 59. What is called an unsound or fallacious Definition of 

an unsound 

argument, that is, an apparent argument, which argument. 

is, in reality, none, cannot, of course, be reduced 

into this form ; but when stated in the form most 

nearly approaching to this that is possible, its when re- 
duced to the 
fallaciousness becomes more evident, from its form, the fat 

r \ r • i t^ ^ acv is more 

nonconformity to the foregoing rule, ror ex- evident. 
ample : 

" Whoever is capable of deliberate crime is responsible ; Example. 

An infant is not capable of deliberate crime ; 
Therefore, an infant is not responsible.'' 

Here the term "responsible" is affirmed uni- Analysis of 
versally of '*' those capable of deliberate crime ;" 
it might, therefore, according to Aristotle's dic- 
tum, have been affirmed of any thing; contained 
under that class ; but, in the instance before us, 
nothing is mentioned as contained under that its defective 
class ; only, the term " infant" is excluded from ^^l 
that class; and though what is affirmed of a 
whole class may be affirmed of anv thing that 
is contained under it, there is no ground for sup- 
posing that it may be denied of whatever is vM 



80 logic. [book I. 



so contained ; for it is evidently possible that it 

the argument may be applicable to a whole class and to some- 

is not good, thing else besides. To say, for example, that all 

trees are vegetables, does not imply that nothing 

else is a vegetable. Nor, when it is said, that 

what the a ]j w j 10 are ca p a |ji e f deliberate crime are re- 
statement ■ 
implies, sponsible, does this imply that no others are 

responsible ; for though this may be very true, 

What is to it has not been asserted in the premiss before us; 

be done in . . 

the analysis *nd in the analysis <>i an argument, we are 
ofan . discard all consideration of what might be 

argument. ° 

serted ; contemplating only what actually is laid 
down in the premises. It is evident, th< 
The one that such an apparent argument as the above 

above did noi j ^ ]y ^ ^ ^ ^y ^ ^ 

comply with * - 

the rule. can j )e so stated as to comply with it, and is 
consequently invalid. 

§ 60. Again, in this install* 

Another " Food is necessary t<> life ; 

example. Com is food ; 

Therefore corn is necessary to life :" 

inwtatthe the term "necessary to life" is affirmed of food, 

argument is 

defective, but not universally ; for it is not said of every 
kind of food the meaning of the assertion be- 
ing manifestly that some food is necessary to 
life : here again, therefore, the rule has not been 
complied with, since that which has been predi- 






CHAP. III.] A N A L Y X It'AL OUTLINE. 81 



cated (that is, affirmed or denied), not of the why we 
whole, but of apart only of a cerlain class, can- ^To^ecm 
not be, on that ground, predicated of whatever w ^ at wa f 

' o 7 I predicated of 

is contained under that class. food - 



DISTRIBUTION AND NON-DISTRIBUTION OF TERMS. 

§ 61. The fallacy in this last case is, what is Fannin the 

last example 

usually described in logical language as consist- 
ing in the "non-distribution of the middle term ;" Non-distnbu. 

tion of the 

that is, its not being employed to denote all the middle term, 
objects to which it is applicable. In order to 
understand this phrase, it is necessary to observe, 
that a term is said to be " distributed," when it is 
taken universally, that is, so as to stand for all 
its significates ; and consequently •'•' undistribu- 
ted/' when it stands for only a portion of its sig- 
nificates.* Thus, " all food," or evert/ kind of what distri- 
food, are expressions which imply the distribu- 
tion of the term "food;" "some food" would Non-distribu- 
jmply its non-distribution. 

Now, it is plain, that if in each premiss a part 
only of the middle term is employed, that is, if 
it be not at all distributed, no conclusion can 

Hoav the ex- 
be drawn. Hence, if in the example formerly ample might 

adduced, it had been merely stated that " some- v aricd . 



* Section 15. 
6 



82 logic. [book I. 

thing' (not " whatever" or " every thing') 

" which exhibits marks of design, is the work of 

an intelligent author," it would not have fol- 

w hat it l owe d f rom the world's exhibiting marks of de- 

woul.l then ° 

have implied, sign, that that is the work of an intelligent author. 
woni«mnrk- § 62. It is to be observed also, that the words 

log distribu- . . . ..... 

tioi.omon- " &H and " every,' which mark the distribution 
\^T.uT ni of a term, and "some," which marks its non- 

IlOl HI \\ <1\ S 

exi.rcs.r.i. distribution, arc not always expressed: they are 
frequently understood, and left to be supplied by 
the context; as. for example, "food is nee 
sary;" viz. * some food ; M "man is mortal;" viz. 

Bachpropo- " every man." Propositions thus expressed are 

nitidis lire t ... 

called called by logicians "indefinite, because it is left 
undetermined by the form of the expression 
whether the subject be distributed or not. Nev- 
ertheless it is plain that in every proposition 
the subject either is or is not meant to be dis- 
tributed, though it be not declared whether 
Dutevcry it is or not ; consequen tly, every proposition, 

proposition . . . 

must be whether expressed indefinitely or not, must be 
either understood as either "universal" or "partfcu- 

Universal or 

rarticuiar. lar ;" those being called universal, in which the 

predicate is said of the whole of the subject 

(or, in other words, where all the significaies 

are included) ; and those particular, in which 

eTcn. 00 only a part of them is included. For example : 






CHAP. III. J ANALYTICAL OUTLINE. 83 

"All men are sinful," is universal: "some men This division 
are sinful," particular ; and this division of prop- _^jj_ 
ositions, having reference to the distribution of 
the subject, is, in logical language, said to be ac- 
cording to their "quantity" 



§ 63. But the distribution or non-distribution Distribution 

of the predi- 

of the predicate is entirely independent of the catenas™ 

reference to 

quantity of the proposition; nor are the signs qua ntit y . 
"all" and "some" ever affixed to the predicate ; 
because its distribution depends upon, and is Hasreference 

to quality, 

indicated by, the " quality" of the proposition ; 
that is, its being affirmative or negative ; it being 
a universal rule, that the predicate of a negative 
proposition is distributed, and of an affirmative, Whenitis 

1 l distributed : 

undistributed. The reason of this may easily 

be understood, by considering that a term which The 

stands for a whole class may be applied to (that 

is, affirmed of) any thing that is comprehended 

under that class, though the term of which it is xnepredicate 

thus affirmed mav be of much narrower extent ° a TOatlve 

propositions 

than that other, and may therefore be far from may be ap " 

plicable to 

coinciding with the whole of it. Thus it may the subject, 
be said with truth, that " the Negroes are unciv- muC h wider 
ilized," though the term " uncivilized" be of much 



reason 
of this. 



wider extent than "Negroes," comprehending, 
besides them, Patagonians, Esquimaux, &c. ; 
so that it would not be allowable to assert, that 



84 logic. [book I 



Hence, only a all who are uncivilized are Negroes/' 5 It is ev- 

term is used. Went, therefore, that it is a part only of the 

term " uncivilized" that has been affirmed of 

" Negroes ; ?; and the same reasoning applies to 

every affirmative proposition. 

Butiti.K.y It may indeed so happen, that the subject 

exJn^with an d predicate coincide, that is, are of equal 

thesubject: extent . aSj f or example : " nil men are rational 

animals ;" " all equilateral triangles arc cquian- 

gular;" (it being equally true, that " all rational 

this not ha- animals are men," and that M all equiangular tri- 
pled in tin- . 

form of the angles are equilateral ; ) yet tins is do! implied 

expression. ^ ^ form of the <\rj>r<ssin/i ; >inrc it Would 

be do less true that "all men are rational ani- 
mals," even if thriv wnv other rational animals 
besides men. 
ifanypartof It is plain, therefore, that if any pari of the 
te applicable Indicate is applicable to the subject, it may be 

tothosub- a flj rnae( i j ;ni( i of course cannot be denied, oi' that 

jeet, it may 

b&Bffinned subject; and consequently, when the predicate 

of the sub- 
ject, is denied of the subject, this implies that n>, 

part of that predicate is applicable to that sub- 
ject ; that is, that the ichole of the predicate is 
ifa predicate denied of the subject: for to say, for example, 
sublet, the that " no beasts of prey ruminate," implies that 
whole predi- j3 easts f prev are excluded from the ichole class 

cate is * y 

denied of f ruminant animals, and consequently that M do 

the subject. 

ruminant animals are beasts of prey."' And 



CHAP. III.] ANALYTICAL OUTLINE. 85 



hence results the above-mentioned rule, that the Distribution 
distribution of the predicate is implied in nega- ^J^edTn 
tive propositions, and its non-distribution in af- ne s ative 

1 l propositions: 

firmativeS. non-distribu- 

tion in 
affirmatives. 

§ 64. It is to be remembered, therefore, that Not sufficient 

for the mid- 
it is not sufficient for the middle term to occur die term to 

, . . . . r , occur in a 

m a universal proposition ; since it that propo- imiver sai 

sition be an affirmative, and the middle term be P r °P° sition - 

the predicate of it, it will not be distributed. 

For example : if in the example formerly given, 

it had been merely asserted, that " all the works 

of an intelligent author show marks of design," 

and that "the universe shows marks of design," u must be so 

nothing could have been proved ; since, though ^^ ^ 

both these propositions are universal, the middle terms of the 

conclusion, 

term is made the predicate in each, and both are that those 

terms may be 

affirmative ; and accordingly, the rule of Aris- compared to- 
totle is not here complied with, since the term 
" work of an intelligent author," which is to be 
proved applicable to " the universe," would not 
have been affirmed of the middle term (" what 
shows marks of design") under which " universe" 
is contained ; but the middle term, on the con- 
trary, would have been affirmed of it. 

If, however, one of the premises be negative, if oneprem- 
the middle term may then be made the predicate 13Sbenega " 



86 LOGIC. [BOOK I. 



the, the mid- of that, and will thus, according to the above 

die term may . . , . . . . ,-, . 

be made the remark, be distributed, r or example : 

predicate of 
that, and will __ . . 

be(listril> " jNo ruminant animals are predacious : 

uted. The lion is predacious ; 

Therefore the lion is not ruminant :'' 



this is a valid syllogism ; and the middle term 

(predacious) is distributed by being made the 

The form of predicate of a negative proposition. The form, 

,hlssyll,> " indeed, of the syllogism is not that prescribed 

gism will not J ° ■ 

iM-ihat pre- \yy \\ lc dictum of Alistotle, l)llt it IlKiV easily be 
scribed by 

the dictum, reduced to that form, by Stating the first prop- 
but may fee ... m T , , 

reduced toit. OSltlon tlins : " JNo predaCH »us animals are ru- 
minant;" which is manifestly implied (as was 
above remarked) in the assertion that '-no ru- 
minant animals are predacious." The syllogism 
will thus appear in the form to which the dictum 
applies. 

aii argu- §65. It is not every argument, indeed, that 

nicnts cannot . . , , . r , , -. 

be reduced can De reduced to this form by so short and sim- 
by so short a j an alteration as in the case before us. A 

process. I 

longer and more complex process will often be 
required, and rules may be laid down to facilitate 
this process in certain cases ; but there is no 
sound argument but what can be reduced into 
But an argu- this form, without at all departing from the real 

men* may meanjng an( j drift f ft . an d the form will be 



CHAP. 111.] ANALYTICAL OUTLINE. 87 



found (though more prolix than is needed for be reduced 
ordinary use) the most per 
argument can be exhibited. 



to the pre- 

ordmary use) the most perspicuous m which an ^-^^ form< 



§ 66. All deductive reasoning whatever, then, aii deductive 

. . , • • i l • i i i reasoning 

rests on the one simple principle laid down by rest 3onthe 
Aristotle., that dictum - 

'•' What is predicated, either affirmatively or 
negatively, of a term distributed, may be predi- 
cated in like manner (that is, affirmatively or neg- 
atively) of any thing contained under that term/' 

So that, when our object is to prove any prop- what are the 
osition, that is, to show that one term mav ri^htlv process * sof 

o » proof. 

be affirmed or denied of another, the process 

which really takes place in our minds is, that we 

refer that term (of which the other is to be thus 

predicated) to some class (that is, middle term) 

of which that other may be affirmed, or denied, 

as the case may be. Whatever the subject-mat- The reason- 
ing always 

ter of an argument may be, the reasoning itself, the same, 
considered by itself, is in every case the same 
process; and if the writers against Logic had Mistakes of 
kept this in mind, they would have been cautious Logic. 
of expressing their contempt of what they call 
" syllogistic reasoning," which embraces all de- 
ductive reasoning; and instead of ridiculing Aris- 
totle's principle for its obviousness and simplicity, Aristotle's 
would have perceived that these are, in fact, its pTinci?le 



88 LOGIC. [book I. 

simple and highest praise : the easiest, shortest, and most 
evident theory, provided it answe] 
of explanation, being ever the best. 



evident theory, provided it answer the purpose 



RULES FOR EXAMINING SYLLOGISMS. 

Testa of the § G7. The following axioms or canons serve 
syllogisms, as tests of the validity of that class of syllo- 
gisms which we have considered, 
ist test. 1 st - If two terms agree with one and the torn* 

third, they tsgree with each other, 
<wtrst. 2d. If one term agree* and another disag 

irif/i one and the tame third, these two disag 
with each of/i •/*. 
The first the Qb the former of these canons rests the Vft- 

test of all ( , . 

affirmative liditv of affirmative conclusions; OD the latter, 

Tftie second (){ MgMfft: for; no syllogism can he faulty 

oTMgattTe. ^jnch dors not violate these canons: none cor- 

rect which does; hence, on these two canODt 

are built the following rules or cautions, which 

are to be observed with respect to syllogisms, 

for the purpose of ascertaining whether those 

canons have been strictly observed or not. 

Every ?yiio- 1st. Every syllogism has three and only three 

gism has ierms . v j z ^ e middle term and the two terms 

three and 

only three f t } ie Conclusion : the terms of the Conclusion 

terms. 

are sometimes called extremes. 
Every pviio- 2d. Every syllogism his three and only thrps 



CHAP. III.] 



ANALYTICAL OUTLINE. 



89 



-propositions; viz. the major premiss; the minor gismhas 

, , , three and 

premiss ; and the conclusion. only three 

3d. If the middle term is ambiguous, there P r °P° sltlons - 

7 . . , 77 . . , Middle term 

are in reality two middle terms, in sense, though must not bv 

but one in sound. ambiguom 

There are two cases of ambiguity: 1st. Where Two cases 

the middle term is equivocal ; that is, when used lst ca3e< 
in different senses in the two premises. For 
example : 



" Light is contrary to darkness ; 
Feathers are light ; therefore, 
Feathers are contrary to darkness." 



Example. 



2d. Where the middle term is not distrib- 
uted ; for as it is then used to stand for a part 
only of its significates, it may happen that one 
of the extremes is compared with one part of 
the whole term, and the other with another part 
of it. For example : 



2d case. 



Again : 



Examples. 



,; White is a color ; 
Black is a color ; therefore, 
Black is white." 

" Some animals are beasts ; 
Some animals are birds ; therefore, 
Some birds are beasts." 



The middle 

3d. The middle term, therefore, must be dis- term must be 

once distrib 

tributed, once, at least, in the premises ; that is, u ted; 



90 



LOGIC. 



[book I. 



and cnce is 
sufficient. 



No term must 
be distribu- 
ted in Die 
conclu-i<>ri 
which was 
not distribu- 
ted in ■ 
premiss. 



Example 



Negative 

premises 

prove noi.i 

intf. 



fcxaro 



by being the subject of a universal,* or predi- 
cate of a negative ;f and once is sufficient ; 
since if one extreme has been compared with a 
part of the middle term, and another to the 
whole of it, they must have been compared with 
the same. 

4th. No term must be distributed in the con- 
clusion, which was not distributed in one of the 
premises j for, that would be to employ the 
whole of a term in the conclusion, when you 
had employed only a pari of it in the premiss; 
thus, in reality, to introduce a fourth term. 
This is called an illicit process cither of the 
major or minor term.} Pot example : 

" All qnadnipecls arc animals, 
A bird is not a quadruped ; therefore, 

It is not an animal." Illicit process of the major. 

5th. From negative premises you can infer 
nothing. For, in them the Middle is pronounced 
to disagree with both extremes ; therefore they 
cannot be compared together : for, the extremes 
can only be compared when the middle agrees 
with both ; or, agrees with one, and disagrees 
with the other. For example : 

" A fish is not a quadruped ;" 

" A bird is not a quadruped," proves nothing. 



* Section 62. f Section 63. X Section 40. 



ANALYTICAL OUTLINE. 91 

6th. If one premiss be negative, the conclu- if oneprem- 

7 . r . . , iss is nega- 

sion must be negative; lor, m that premiss the t ive, the 



middle term is pronounced to disagree with one c " nclusl0n 

1 will be pega- 

of the extremes, and in the other premiss (which tive ; 
of course is affirmative by the preceding rule), 
to agree with the other extreme ; therefore, the 
extremes disagreeing with each other, the con- 
clusion is negative. In the same manner it may andrecipro. 
be shown, that to prove a negative conclusion, 
one of the premises must be a negative. 

By these six rules all Syllogisms are to be wnatfoi- 

• i i r i -n i -i lows from 

tried; and from them it will be evident, 1st, these six 
that nothing can be proved from two particular 
premises; (since you will then have either the 
middle term undistributed, or an illicit process. 
For example : 



rules. 



" Some animals are sagacious ; 
Some beasts are not sagacious ; 
Some beasts are not animals.") 

And, for the same reason, 2dly, that if one of 2d inference, 
the premises be particular, the conclusion must 
be particular. For example : 



" All who fight bravely deserve reward ; 

" Some soldiers fight bravely ;" you can only infer that 

" Some soldiers deserve reward :" 

for to infer a universal conclusion would be 
an illicit process of the minor. But from two 



Example. 



\)2 LOGIC. [BOOK I. 



rwouniver- universal Premises you cannot always infer a 

sal premises . , ~ , -^ . 

do not always universal Conclusion, ror example: 

give a uni- 
versal con- " All gold is precious ; 

All gold is a mineral ; therefore, 

Some mineral is precious.' 



elusion. 



And even when we can infer a universal, we 
are always at liberty to infer a particular ; since 
what is predicated of all may of course be pre 
dicated of some. 



OF FALLACIES. 

Definition of § 08. By a fallacy is commonly understood 

a fallacy. 

"any unsound mode of arguing, which appears 

to demand our conviction, and to be decis 

of the question in hand, when in fairness it is 

notation of, not/' In the practical detection of each indi- 

acuteness. vidual fallacy, much must depend on natural 

and acquired acuteness ; nor can any rules be 

given, the mere learning of which will enable 

us to apply them with mechanical certainty and 

Hints and readiness ; but still we may give some hints that 

will lead to correct general views of the subject, 

and tend to engender such a habit of mind, as 

will lead to critical examinations. 

same of lo- Indeed, the case is the same with respect to 

gicingenerni. ^ g lQ/ i n general; scarcely any one would, in 

ordinary practice., state to himself either his 



CHAP. III.] ANALYTICAL OUTLINE. 93 



own or another's reasoning, in syllogisms at full Logic tends 

length ; yet a familiarity with logical principles habitj of 

tends Very much (as all feel, who are really well clear J^ a30n - 

acquainted with them) to beget a habit of clear 

and sound reasoning. The truth is, in this as 

in many other things, there are processes going The habit 

fixed, we 

on in the mind (when we are practising any naturally foi- 
thing quite familiar to us), with such rapidity proce5ses . 
as to leave no trace in the memory ; and we 
often apply principles which did not, as far as 
we are conscious, even occur to us at the time 



§ 69. Let it be remembered, that in every conclusion 

r . l __f n .it follows from 

process of reasoning, logically stated, the con- twoantece _ 
elusion is inferred from two antecedent propo- deut prem " 

ises. 

sitions, called the Premises. Hence, it is man- 
ifest, that in every argument, the fault, if there Fallacy, if 

-I - 1 . - any, either in 

be any, must be either, {he premises 

1st. In the premises; or, 
2d. In the conclusion (when it does not follow orconciu- 

r . N sion, or both. 

Irom them) ; or, 

3d. In both. 

In every fallacy, the conclusion either does or 
does not follow from the premises. 

When the fault is in the premises; that is, when m the 
when they are such as ought not to have been prerai9es; 
assumed, and the conclusion legitimately follows 
from them, the fallacy 's called a Material Fal- 



conclusion. 



94 LOGIC. [book I. 

lacy, because it lies in the matter of the argu- 
ment, 
when in the Where the conclusion does not follow from 
the premises, it is manifest that the fault is in 
the reasoning, and in that alone: these, there- 
fore, are called Logical Fallacies, as being prop 
erly violations of those rules of reasoning which 
it is the province of logic to lay down. 

When the fault lies in both the premises and 
reasoning the fallacy is both Material and I < 



W T hen in 
both. 



Rules for 
Dx;uninin'4 
train of ar- 
gument. 

1st Rule. 



§70. In examining a train of argumentation, 

examining. ascerta j u if B hillaev have eivpt into it, the 

train of ar- 
gument, following points would naturally suggest them* 

selves ; 

1st. What is the proposition to he proi 
On what facts or truths, as premises, is the ar- 
gument to rest? and. What are the marks 
truth by which the Conclusion may he known? 

2d. Are the premises both true? If facts, are 
they substantiated by sufficient proofs ? If truths, 
were they logically inferred, and from correct 
premises ? 

3d. Is the middle term what it should be, and 
the conclusion logically inferred from the prem- 
ises ? 

These general suggestions may serve as guides 
in examining arguments for the purpose of de- 



2d Rule. 



3d Rule. 



Suggestions 
serve as 
guides, 



CHAP. III.] ANALYTICAL OUTLINE. 95 

tecting fallacies; but however perfect general to detect 

. , . error. 

rules may be, it is quite certain that error, m 
its thousand forms, will not always be separated 
from truth, even by those who most thoroughly 
understand and carefully apply such rules 



CONCLUDING RE 31 ARKS. 

§ 71. The imperfect and irregular sketch which Logic 

corresponds 

has here been attempted of deductive logic, may with tbe 

reasonings in 
Geometry. 



suffice to point out the general drift and purpose 
of the science, and to show its entire correspond- 
ence with the reasonings m Geometrv. The 
analytical form, which has here been adopted, Analytical 
is, generally speaking, better suited for introdic- form * 
cing any science in the plainest and most inter- 
esting form ; though the synthetical is the more syntheiieai 
regular, and the more compendious form for sto- 
ring it up in the memory. 

§ 72. It has been a matter about which wri- induction: 

. does it form 

ters on logic have dmered, whether, and in con- apart f 
formity to what principles, Induction forms a Lo ^ c - 
part of the science ; Archbishop Whately main- wnateiy- 3 
taining that logic is only concerned in inferring 
truths from known and admitted premises, and 
that all reasoning, whether Inductive or Deduc- 
tive, is shown by analysis to have the syllogism 



opinion 



9G logic. [book I. 

Murs views, for its type ; while Mr. Mill, a writer of perhaps 
greater authority, holds that deductive logic is 
but the carrying out of what induction begins ; 
that all reasoning is founded on principles of in- 
ference ulterior to the syllogism, and that the 
syllogism is the test of deduction only. 

Without presuming at all to decide defini- 
tively a question which has been considered and 

Reasons for passed upon by two of the most acute minds of 

the course 

taken. the age, it may perhaps not be out of place to 
state the reasons which induced me to adopt 
the opinions of Mr. Mill in view of the par? 
tfcular life which 1 wished to make offegi 

r filin g ob- § 73. It was, as stated in the g&neral plan, 

jects of tlu* • i %• i 

plan- one of my leading object- to point out the cor- 
respondence between the science of logic and 
the science of mathematics : to show, in fact, 
Tushowthat that mathematical reasoning conforms, in every 
JT^Loning aspect, to the strictest rules of logic, and is in- 
conforms to j^ k ut ] ' c a ppij e j t0 t j ie abstract quantit 

logical rules. Oil 

Number and Space. In treating of space, about 
which the science erf Geometry is conversant, we 
shall see that the reasoning rests mainly on the 
Axioms, how axioms, and that these are established by induc- 
estabhshed. ^ Q processes. The processes of reasoning which 
relate to numbers, whether the numbers are rep- 
resented by figures or letters, consist of two parts- 



CHAP. III.] ANALYTICAL OUTLINE. 97 

1st. To obtain formulas for, that is, to express 
in the language of science, the relations between 
the quantities, facts, truths or principles, what- Two p** 3 of 

the reasoning 

ever they may be, that form the subject of the process. 
reasoning ; and, 

2dly. To deduce from these, by processes 
purely logical, all the truths which are implied 
in them, as premises. 



§ 74. Before dismissing the subject, it may Auinauc- 
be w r ell to remark, that every induction may thrown mo 



the form 
of the 



be thrown into the form of a syllogism, by sup- 
plying the major premiss. If this be done, we Syllogism, by 
shall see that something equivalent to the uni- proper major 
formity of the course of nature will appear as pr 
the ultimate major premiss of all inductions ; 
and will, therefore, stand to all inductions in 
the relation in which, as has been shown, the 
major premiss of a syllogism always stands to 
the conclusion ; not contributing at all to prove 
it, but being a necessary condition of its being 
proved. This fact sustains the view taken by 
Mr. Mill, as stated above ; for, this ultimate ma- How this 

-, . . f. . . r major prem 

jor premiss, or any substitution tor it, is an inter- i^ obtain- 
ence by Induction, but cannot be arrived at by **■ 
means of a svllorism. 



BOOK II. 

MATHEMATICAL SCIENCE. 



CHAPTER I 



QUANTITY AND MATHEMATICAL SCIENCE DEFINED — DIFFERENT KINDS OF QUAN- 
TITY LANGUAGE OF MATHEMATICS EXPLAINED SUBJECTS CLASSIFIED UNIX 

OF MEASURE DEFINED— MATHEMATICS A DEDUCTIVE SCIENCE. 

QUANTITY. 

§ 75. Quantity is a general term applicable Quantity 
to every thing which can be increased or dimin- 
ished, and measured. There are two kinds of 
quantity : 

1st. Abstract Quantity, or quantity, the con- Abstract, 
ception of which does not involve the idea of 
matter; and, 

2dly. Concrete Quantity, which embraces concrete, 
every thing that is material. 



§ 76. Mathematics is the science of quantity ; Mathematics 
that is, the science which treats of the measures 
of quantities and their relations to each other. 
It is divided into two parts : 



100 MATHEMATICAL SCIENCE. [BOOK II 

Pare 1st. The Pure Mathematics, embracing the 

Mathematics. . . , r . . in i 

principles ot the science, and all explanations 
of the processes by which those principles are 
derived from the laws of the abstract quantities, 
Number and Space ; and, 
Mixed 2d. The Mixed Mathematics, embracing the 

' applications of those principles to all investiga- 
tions and to the solution of all questions of a 
practical nature, whether they relate to abstract 
or concrete quantity. 

Mathematics, § 77. Mat hen iat ics, in its primary signifiea- 

. as used by . . . . - - 

ihe ancients: tlon > as US(,( l by the ancients, embraced every 
acquired science, and was equally applicable to 
all branches of knowledge. Subsequently it was 
restricted lo those branches only which v. 
acquired by severe study, or discipline, and its 

embraced an votaries were called Disciples. Those subj. 

subjects . -, i • i -i • • • 

which were therefore, wliieli required patient investigation, 
disciplinary exact rcason i niJ r, :m d the aid of the matlieinati- 

in their na- ° 

turc - cal analysis, were called Disciplinal or Mathe- 
matical, because of the greater evidence in the 
arguments, the infallible certainty of the conclu 
sions, and the mental training and development 
which such exercises produced. 



Pure § 78. It has already been observed that the 

a lcmatios, ^ uve Mathematics embrace all the principles of 



CHAP. I.] NUMBER. 101 



the science, and that these principles are de- what they 

duced, by processes of reasoning upon the two relate to 

abstract quantities, Number and Space. All "^^ 

the definitions and axioms, and all the truths 

deduced from them, are traceable to those two 

sources. Here, then, two important questions Two ques- 
tions, 
present themselves : 

1st. How are we to attain a clear and true Howdowe 

conceive of 

conception of these quantities ? and, tne <i ujmti - 

2dly. How are we to represent them, and what How repre- 
language are we to employ, so as to make their Knt them * 
properties and relations subjects of investiga- 
tion ? 

N U M B E R . 

§ 79. Numbers are expressions for one or Nuinber 
more things of the same kind. How do we denned, 
attain unto the significations of such expres- Howweob . 
sions ? By first presenting to the mind, through tain m idea 

J r & ° of number. 

the eye, a single thing, and calling it one. 
Then presenting two things, and naming them 
two : then three things, and naming them three ; 
and so on for other numbers. Thus, we acquire 
primarily, in a concrete form, our elementary It is done by 
notions of number, by perception, comparison, P erce P tl0U ' 

7 J a a » r - comparison, 

and reflection ; for, we must first perceive how aDd 

reflection. 

many things are numbered ; then compare what 

is designated by the word one, with what is Reason 



102 



MATHEMATICAL SCIENCE. 



[BOOK II. 



iesignated by the words two, three, &c, and 
then reflect on the results of such comparisons 
until we clearly apprehend the difference in the 
signification of the words. Having thus acquired, 
in a concrete form, our conceptions of numbers, 
we can consider numbers as separated from any 
particular objects, and thus form a conception 
Two axioms of them in the abstract. We require but two 
the formation axioms for the formation of all numbers : 

of numbers, j^ j^ (mQ mQy be ^^ {Q ^ 11Um ber, 

1st axiom, and that the number which results will be great- 
er by one than the number to which the one 
was added. 

2d axiom. 2d. That one may be divided into any num- 
ber of equal parts. 



Language 
employed. 



The ten 



§ 80. But what language are we to employ 
as best suited to furnish instruments of thought, 
and the means of recording our ideas and ex- 
pressing them to others ? The ten characters, 

XhXr S called fl S ures > are the ^lp habet °f this language, 
and the various ways in which they are com- 
bined will be fully explained under the head 
Arithmetic, a chapter devoted to the considera- 
tion of numbers, their laws and language. 



d 

■ 



CHAP. I.] 



SPACE. 



103 



SPACE. 



§ 81. Space is indefinite extension. We ac- **** 

defined. 

quire our ideas of it by observing that parts of 
it are occupied by matter or bodies. This ena- 
bles us to attach a definite idea to the word 
place. We are then able to say, intelligibly, Place; 
that a point is that which has place, or position a ^ ohlU 
in space, without occupying any part of it. Hav- 
ing conceived a second point in space, we can 
understand the important axiom, " A straight 
line is the shortest distance between two points ;" Axiom con- 
and this line we call length or a dimension of str aight line, 
space. 



Breadth 
defined. 



§ 82. If we conceive a second straight line 
to be drawn, meeting the first, but lying in a 
direction directly from it, we shall have a second 
dimension of space, which we call breadth. If 
these lines be prolonged, in both directions, they 
will include four portions of space, which make 
up what is called a plane surface or plane : 
hence, a plane has two dimensions, length and 
breadth. If now we draw a line on either side 
of this plane, we shall have another dimension of 
space, called thickness: hence, space has three space has 
dimensions — length, breadth, and thickness. 



A plane 
defined. 



three dimen 
eiona 



104 MATHEM A T I C A L SCIE N G E . [BOOK II 

Figure § 83. A portion of space limited by bounda- 

ries, is called a Figure. If such portion of space 
Line defined have but one dimension, it is called a line, and 
may be limited by two points, one at each ex- 
Two kinds of tremity. There are two kinds of lines, straight 
straight and an & curved. A straight line, is one which does 
curved. no ^ c h anS r e its direction between any two of its 
points, and a curved line constantly changes its 
direction at every point. 



Surface : 



§81. A portion of space having tWO dimen- 
sions is called a surface. There are two kinds 

Plane, 

Curved, of surfaces — Plane Surfaces and Curved Sur- 
faces. With the former, a straight line, hai 

Difference. & 

tWO points ill Common, will always coincide, 

however it mav be placed, while with the latter 

Boundaries 

of a surface, it will not. The b<>undari< are 

lines, straight or curved. 

§ 85. A portion of spade having three dimen- 
sions, is called a solid, and solids are bounded 
either by plane or curved surfaces. 



6olid defined. 



§ 86. The definitions and axioms relating to 
space, and all the reasonings founded on them, 
science of make up the science of Geometry. They will 
all be fullv treated under that head. 



CHAP. I.] 



ANALYSIS. 



10c 



ANALYSIS. 



§ 87. Analysis is a general term embracing Analyst 
all the operations which can be performed on 
quantities when represented by letters. In this 
branch of mathematics, all the quantities con- 
sidered, whether abstract or concrete, are rep- Quantities 
resented by letters of the alphabet, and the 
operations to be performed on them are indi- 
cated by a few arbitrary signs. The letters 
and signs are called Symbols, and by their com- 
bination we form the Algebraic Notation and 
Language. 



represented 
by letters. 



Symbols. 



§ 88. Analysis, in its simplest form, takes the ******* 

* J r Algebra ; 

name of Algebra ; Analytical Geometry, the Dif- Analytical 
ferential and Integral Calculus, extended to in- 
clude the Theory of Variations, are its higher 
and most advanced branches. 



§ 89. The term Analysis has also another sig- Term AmI) 

.sis defined. 

nification. It denotes the process of separating 
any complex whole into the elements of which its nature. 
it is composed. It is opposed to Synthesis, a 
term which denotes the processes of first con- 
sidering the elements separately, then combining 
them, and ascertaining the results of the combi- 
nation. 



Synthesis 
defined. 



100 



MATHEMATICAL SCIENCE. [BOOK II. 



Analytical The Analytical method is best adapted to in- 
racthod ' vestigation, and the presentation of subjects in 
synthetical their general outlines ; the Synthetical method 
method. . g best a dapted to instruction, because A exhib- 
its all the parts of a subject separately, and in 
their proper order and connection. Analysis 
deduces all the parts from a Avhole : Synth 
forms a whole from the separate parts. 

Arithmetic, § 90. Arithmetic, Algebra, and Geometry are 
ciSS, the elementary branches of Mathematical Bci* 

r,r,m, " ti,ry once Every truth which IS established by 

bninchcs. v ' J 

mathematical reasoning, is developed by an 
arithmetical, geometrical, or analytical proc 
or by a combination of them. The reasoning 

in eacb branch is conducted mi principles iden- 
tically the same. Every sign, or symbol, or 
technical word, is accurately defined, so that to 
each there is attached a definite and precise 
j-anguage idea. Thus, the language is made so exact and 
exacl " certain, as to admit of no ambiguity. 



LANGUAGE OF MATHEMATICS. 

Language of § 91. The language of Mathematics is mixed. 

m m^d! 1CS Although composed mainly of symbols, which 
are defined with reference to the uses which 
are made of them, and therefore have a pre 



CHAP. I.] LANGUAGE OF MATHEMATICS. 1 07 

cise signification ; it is also composed, in part, 
of words transferred from our common language. 
The symbols, although arbitrary signs, are, nev- symbols 

general. 

ertheless, entirely general, as signs and instru- 
ments of thought ; and when the sense in which 
they are used is once fixed, by definition, they 
preserve throughout the entire analysis precise- 
ly the same signification. The meaning of the words bor- 

. , , r 1 , rowed from 

words borrowed irom our common vocabulary is comm0 n 
often modified, and sometimes entirelv changed, lan ^ ua ^ 

J ° are modified 

when the words are transferred to the language and used in a 

technical 

of science. They are then used in a particular sense, 
sense, and are said to have a technical significa- 
tion. 



§ 92. It is of the first importance that the Language 

must be 
understood : 



elements of the language be clearly understood, 



— that the signification of every word or sym- 
bol be distinctly apprehended, and that the con- 
nection between the thought and the word or 
symbol which expresses it be so well established 
that the one shall immediately suggest the other. 
It is not possible to pursue the subtle reasonings Mathemati- 
of Mathematics, and to carry out the trains of ^ require 
thought to which they give rise, without entire u * 
familiarity with those means which the mind 
employs to aid its investigations. The child cannot use 

any language 

cannot read till he has learned the alphabet ; 



108 MATHEMATICAL SCIENCE. [booKII. 

well tm we nor can the scholar feel the delicate beauties of 
Shakspeare, or be moved by the sublimity of 
Milton, before studying and learning the Ian 
guage in which their immortal thoughts are 
clothed. 

Quantity § 93. All Quantities, whether abstract or r 

are repre- 
sented by crete, are, in mathematical science, presented 

ajrimoper- to tne &&&& by arbitrary symbols. They arc 4 

ntedonby v j ( > W( . ( | ., !l( [ Qp$y ft ted on tbTOUgh these SVIllhols 

boin. which represent them; and all operations are 
indicated by another cl;iss of symbols calied 
Hgna. si^Ns. The-: ,:>ined with the symi> 

which represent the quantities, make up. 
totofthe vve have slated above, the pure mathematical 

language. 

language; and this, in connection with that 
which is hoi-rowed from our common lanuu 
forms the language "l mathematical sciei. 
This lanun-'iu'e is at once comprehensive and 
its nature, accurate. It is capable of stating the most 
general proposition, and presenting to the mind. 
in their proper order, every elementary princi- 
vvimtitao- pie connected with its solution. By its gener- 

complishes. a]ity it reaches Qver the who ] e fie ] d f the 

pure and mixed sciences, and gathers into con- 
densed forms all the conditions and relations 
necessary to the development of particular ft 
and universal truths; and thus, the skill o( the 



CHAP. I.] 



QUANTITY MEASURED. 



109 



analyst deduces from the same equation the ve- Extent and 
locity of an apple falling to the ground, and the Analysis 
verification of the law of universal gravitation. 

Q U A N T I T Y M BASURED. 



§ 94. Quantity has been defined, * any thing 
which can be increased or diminished, and meas- 
ured.'"' The terms increased or diminished, are 
easily understood, implying merely the property 
of being made larger or smaller. The term 
measured is not so easily explained, because it 
has only a relative meaning. 

The term " measured," applied to a quantity, 
implies the existence of some known quantity 
of the same kind, which is regarded as a stand- 
ard, and with which the quantity to be meas- 
ured is compared with respect to its extent or 
magnitude. To such standard, whatever it may 
be, we give the name of unity, or unit of meas- 
ure ; and the number of times which any quan- 
tity contains its unit of measure, is the numerical 
value of the quantity measured. The extent 
or magnitude of a quantity is, therefore, merely 
relative, and hence, we can form no idea of it, 
except by the aid of comparison. Space, for 
example, is entirely indefinite, and we measure 
parts of it by means of certain standards, called 



Quantity. 



Increased 

and 

diminished, 

defined. 



Measured, 



What It 
means. 



Standard : 

is called 
unity. 



Magnitude : 
merely rela- 
tive. 



fcpace : 
indefinite* 



110 MATHEMATICAL SCIENCE. [ B00K * 



Measurement measures ; and after any measurement is com- 

ancertains re- . . . 

jaiion: pletecl, we have only ascertained the rekUum or 

proportion which exists between the standard we 

a pr««» of adopted and the thing measured. Hence, measure- 

comparisoiL 

ment is, after all, but a mere process of comparison. 

wv.-i.t and § 1)5. The abstract quantities, Weight 

known by Velocity, are but vague and indefinite CMC 
comparison. t j 011Sj un tji compared with their units erf m< 

ure, mid even these are arrived at only by pro- 
Compahson cesses of comparison. Indeed, most of oar 

,,„.,,, imU knowledge Of all subjects is obtained in the 

same way. We compare together, very care- 
fully, all the facts which form the basis of an 
induction; and we rely on the comparison of 
the terms in the major and minor premises for 
every conclusion by a deductive prOC< 

Quiniiiy. § 06. Quantity, as we have seen, is divided 

into Abstract and Concrete — the abstract quan- 
Ai.str.Kt. tity being a mere mental conception, having 
for its sign a number, a letter, or a ge om etric a l 
concrete, figure. A concrete quantity is a physical ob- 
ject, or a collection of such objects, and may 
now repre- likewise be represented by numbers, letters, or 
aent by th e geometrical magnitudes regarded as ma- 

Exampieof terial. The number "three" is entirely abstract, 

the abstract. , , ... 

expressing an idea having no connection with 






CHAP. I.] PURE MATHEMATICS. Ill 

material things ; while the number " three pounds 
of tea," or " three apples/' presents to the mind 
an idea of physical objects. So, a portion of Exam r ,leof 

r J J L the abstract 

space, bounded by a surface, all the points of 
which are equally distant from a certain point 
within called the centre, is but a mental con- 
ception of form; but regarded as a solid mass, of the con- 
crete. 
it gives rise to the additional idea of a material 

substance. 

PURE MATHEMATICS. 

§ 97. The Pure Mathematics are based on Pure 

Mathematics: 

definitions and intuitive truths, called axioms, 

which are inferred from observation and expe- what are its 

, . , . . foundations. 

rience ; that is, observation and experience fur- 
nish the information necessary to such intuitive 
inductions.* From these definitions and axioms, 
as premises, all the truths of the science are estab- 
lished by processes of deductive reasoning ; and 
there is not, in the whole range of mathemat- Itste8tsof 

J truths: 

ical science any logical test of truth, but in a 
conformity of the conclusions to the definitions what tbey 
and axioms, or to such principles as have been 
established from them. Hence, we see, that in what the 
the science of Pure Mathematics, which con- 
sists merely in inferring, by fixed rules, all the 



science con- 
sists. 



* Section 27. 



112 MATHEMATICAL SCIENCE. [BOOK II. 



is purely truths which can be deduced from given prem- 

Deductive. . . , r\ j *• c? • rru 

ises, is purely a Deductive ocience. 1 he pre- 
cision and accuracy of the definitions ; the cer- 
tainty which is felt in the truth of the axioms ; 
Precision of the obvious and fixed relation between the I 

its language. 

and the thing signified; and the certain for- 
mulas to which the reasoning processes are re- 
duced, have giron to mathematics the name of 

Exact 
Science. " Exact Science." 



AiirensoninK § 98. We have remarked thai all the reaaon- 
' iii«js of mathematical science, and all tin* truths 

initions bub 

axioms. which they establish, are based on the defini- 
tions and axioms which correspond to the major 
premiss of the SyHoglSftl. If the resemblance 
which the minor premise asserts to the middle 
Relations not term were ol)\ioiis to the leases, as it is in the 

obvious. 

proposition. "Socrates was a man. OT W< 

at once ascertainable by direct observation, of 

were as evident as the intuitive truth. "A whole 

is equal to the sum of all its parts :" there 

Deductive would be no necessity for trains of reasoning, 

nece nary, and Deductive Science would not exist. Trains 

Trains of of reasoning are necessary only for the sake of 

reasonm-: extem |j n g t ) ie definitions and axioms to other 

what they cases in which we not only cannot directly ob- 

BC(,ompis . serye w ] iat j s | b e proved, but cannot directly 

observe even the mark which is to prove it. 



CHAP. I.] PURE MATHEMATICS. 113 



§ 99. Although the syllogism is the ultimate syllogism, 

,, , , . . /l'li- the final test 

test in all deductive reasoning (and indeed in f deduction, 
all inductive, if we admit the uniformity of the 
course of nature), still we do not find it con- 
venient or necessary, in mathematics, to throw 
every proposition into the form of a syllogism. 

The definitions and axioms, and the propo- Axioms and 
sitions established from them, are our tests of tests of truth: 
truth; and whenever any new proposition can 

be brought to conform to any one of these a proposi- 
tion: when 
tests, it is regarded as proved, and declared to proved. 

be true. 



§ 100. When general formulas have been When a 

principle 

framed, determining the limits within which the may be re- 
deductions may be drawn (that is, what shall ^tlt™ 
be the tests of truth), as often as a new case 
nan be at once seen to come within one of the 
formulas, the principle applies to the new case, 
and the business is ended. But new cases are Trains of 
continually arising, which do not obviously come reasomng: 

^ ° J why neces- 

within any formula that will settle the questions sar y- 
we want solved in regard to them, and it is 
necessary to reduce them to such formulas. 
This gives rise to the existence of the science ti,.. 
of mathematics, requiring the highest scientific nsetolh " 

A ° ° science of 

genius in^ those who contributed to its creation, mathematics, 
and calling for a most continued and vigorous 

6 



114 MATHEMATICAL SCIENCE. [BOOK II. 

exertion of intellect, in order to appropriate it, 
when created 



COMPARISON OF QUANTITIES. 

Mathematics § 101 yf e have seen that the pure mathe- 

concerned 

withNumhor matics are concerned with the two abstract 

► pace. q Uallt j t j rs \unihcr and Space. We have 
Roasonm- seen that reasoning ffily involves coin- 

involves . i • i 

compah^n. parison : hence, mat hemal ical reasoning must 
consist in comparing the quantitiea which come 

from Number and Space with each other 



Twoquunti- § io2, Any two quantities, compared with 

tics c:m sus- 
tain but two each other, must necessarily sustain one of two 

relations: they must be equal at unequal. What 

axioms or formulas have we for inferring the 

one or the other ? 



AXIOMS OK FORMULAS FOR INFERRING EQUALITY. 

1. Things which being applied to each other 
coincide, are equal to one another. 

Formulas 2 . Things which are equal to the same thing 

for ° l ° 

Equality, are equal to one another. 

3. A whole is equal to the sum of all its parts. 

4. If equals be added to equals, the sums are 
**nual. 



CHAP, ij COMPARISON OF aUANTITIES. 



115 



5. If equals be taken from equals, the remain- 
ders are equal. 

AXIOMS OR FORMULAS FOR INFERRING INEQUALITY. 

1. A whole is greater than any of its parts. 

2. If equals be added to unequals, the sums Formulas 
are unequal. . {oT ,. t 

^ Inequality. 

3. If equals be taken from unequals, the re- 
mainders are unequal. 



§ 103. We have thus completed a very brief omiineof 
and general analytical view of Mathematical '^^ e 7 
Science. We have endeavored to point out 
the character of the definitions, and the sources 
as well as the nature of the elementary and in- 
tuitive propositions on which the science rests ; What fea- 
the kind of reasoning employed in its creation, heen 
and its divisions resulting from the use of dif- * ketched - 
ferent symbols and differences of language. We 
shall now proceed to treat the subjects separ- 
ately. 



CHAP. II.] ARITHMETIC FIRST NOTIONS. 117 



CHAPTER II. 



ARITHMETIC SCIENCE AND ART OF NUMBERS. 



SECTION I. 



INTEGER UNITS. 



FIRST NOTIONS OF NUMBERS. 

§ 104. There is but a single elementary idea Rut one eie- 

. r mentary idea 

in the science of numbers : it is the idea of the in numbers. 
unit one. There is but one way of impressing Howim- 
this idea on the mind. It is by presenting to P t hTmind! 
the senses a single object ; as, one apple, one 
peach one pear, &c 



§ 105. There are three signs by means of Three signs 

. for express- 

which the idea of one is expressed and commu- ingit. 
aicated. They are, 

1st. The word one. a word. 

2d. The Roman character I. Roman 

__, _ character: 

3d. The figure 1. 

° Figure. 



118 MATHEMATICAL SCIENCE [ B00K n 



New ideas § 106. If one be added to one, the idea thus 

which arise .....—. r , 

by adding arising is different from the idea ol one, and is 
complex. This new idea has also three signs ; 
viz. two, II., and 2. If one be again added, 
that is, added to two, the new idea has likewise 
three signs; viz. three, III., and 3. The ex- 

Theexpres- pressions for these, and similar ideas, are called 

eions aro 

numbers, numbers : hence, 

Numbers Numbers are expressions for one Of more 
things of the tame hind. 



IDEAS OF NUMBERS GENERALIZED. 

ideas of § 107. If we begin with the idea of the num- 

genenUized. Der 0nC > ail( l t ' 1011 a( ' ( ' li t() n,ie - Dttakillg tWO J 

and then add it to two, making three ; and then 

to three, making four; and then to four, making 
How formed, five, and so on ; it is plain that we shall form a 

series of numbers, each of which will be giemtef 
unity iho by one than that which precedes it. Now, one 

or unity, is the basis of this series of numb' 
of expressing and each number may be expressed in three 

them. 

ways : 
1st way. 1st. By the words one, two, three, &c, of oui 

common language ; 
2d way. 2d. By the Roman characters ; and, 

3d way. 3d. By figures. 



CHAP. II.] ARITHMETIC UNITY. 119 



notions are 
complex. 



§ 108. Since all numbers, whether integer or ah numbers 
fractional, must come from, and hence be con- one . 
nected with, the unit one, it follows that there 
is but one purely elementary idea in the science 
of numbers. Hence, the idea of every number, Hence but 
regarded as made up of units (and all numbers iapureiyeie-* 
except one must be so regarded when we ana- mentar >- 
lyze them), is necessarily complex. For, since au other 
the number arises from the addition of ones, the 
apprehension of it is incomplete until we under- 
stand how those additions were made ; and there- 
fore, a full idea of the number is necessarily 
complex. 

§ 109. But if we regard a number as an en- 
tirety, that is, as an entire or whole thing, as an 
entire two, or three, or four, without pausing to when a 
analyze the units of which it is made up, it may ^^garded* 
then be regarded as a simple or incomplex idea ; asmcom P Iex - 
though, as w r e have seen, such idea may always 
be traced to that of the unit one, which forms 
the basis of the number. 



UNITY AND A UNIT DEFINED. 

§ 110. When we name a number, as twenty whatisne- 
feet, two things are necessary to its clear appre- cessar y tothe 

J *■ l apprehension 

hension. of a number 



120 MATHEMATICAL SCIENCE. [BOOK II. 



UNIT. 



First. 1st. A distinct apprehension of the si?igle 

thing which forms the basis of the number ; and, 

socond. 2d. A distinct apprehension of the number of 

times which that thing is taken. 

The basis of The single thing, which forms the basis of the 

the number , n j t a • 11 1 

is unity, number, is called unity, or a unit. It is called 
when it is unity, when it is regarded as the prinidnj b</sis 
called unity, f t h e num ber; that is, when it is the final stand- 
ard to which all the numbers that come from it 
andwhnwi MRS referred. It is called a unit when it is re- 
garded as one of the collection of several equal 
things which form a number. Thus, in the ex- 
ample, one foot, regarded ;is ;t standard and the 
baas of the Dumber, is called i'mtv ; hut. con- 
sidered as one of the twenty equal feel which 
make up the number, it is called a UNIT. 



OF SIMPLE AND DENOMINATE NUMBERS. 

Abstract § HI- A simple or abstract unit, is <>\i:. with- 

out regard to the kind of thing to which the term 
one may be applied. 

A denominate or concrete unit, is one thing 
named or denominated ; as, one apple, one peach, 
one pear, one horse, &c. 



unit. 



Denominate 
unit. 



Number has § 112. Number, as such, has no reference 
to the particular things numbered. But to dis- 



CHAP. II.] ARITHMETIC ALPHABET. 121 

tinguish numbers which are applied to particular to the thing* 

units from those which are purely abstract, we 

call the latter Abstract or Simple Numbers, simple 

and 

and the former Concrete or Denominate Xum- Denominate. 
bers. Thus, fifteen is an abstract or simple 
number, because the unit is one; and fifteen Example*, 
pounds is a concrete or denominate number, 
because its unit, one pound, is denominated or 
named. 



ALPHABET WORDS GRAMMAR. 

§ 113. The term alphabet, in its most general Alphabet, 
sense, denotes a set of characters which form 
the elements of a written language. 

When any one of these characters, or any VVords - 
combination of them, is used as the sign of a 
distinct notion or idea, it is called a word ; and 
the naming of the characters of which the word 
is composed, is called its spelling. 

Grammar, as a science, treats of the estab- Grammar 
lished connection between words as the signs of 
ideas. 

ARITHMETICAL ALPHABET. 

§114. The arithmetical alphabet consists of ArithmeUc-i 
ten characters, called figures. They are, 

Naught, One, Two, Three, Fcur, Five, Six, Seven, Eight, Nine, 

012 3456789 



122 MATHEMATICAL SCIENCE. [BOOK II. 

and each may be regarded as a word, since it 
stands for a distinct idea. 



WORDS SPELLING AXD READING IN ADDITION. 

one cannot § 115. The idea of one, being elementary, the 
espe ' character 1 which represents it, is also element- 
ary, and hence, cannot be spelled by the other 
characters of the Arithmetical Alphabet (§ 114). 
But the idea which is expressed by 2 comes from 

spelling by the addition of 1 and 1 : hence, the word repre- 

the 

arithmetical sented by the character 2, may be spelled by 

characters, j ^ j rp^ j ^j j ^ ^ ^ ^ ^fa^ 

ical spelling of the word two. 

Three is spelled thus: 1 and 2 are 3; and 
also, 2 and 1 are 3. 
Examples. ]? our j s spelled, \ a nd 3 are 4 ; 3 and 1 are 4 ; 
2 and 2 are 4. 

Five is spelled, 1 and 4 are 5 ; 4 and 1 are 5 ; 
2 and 3 are 5 ; 3 and 2 are 5. 

Six is spelled, 1 and 5 are G ; 5 and 1 are 6 ; 
2 and 4 are 6 ; 4 and 2 are G ; 3 and 3 are 6. 

Ail numbers § 116. In a similar manner, any number in 
died hi a arithmetic may be spelled; and hence we see 

similar way. t [ iat ^q p rocess f spelling in addition consists 
simply, in naming any two elements which will 
make up the number. All the numbers in ad 






CHAP II. J ARITHMETIC READINGS. 



123 



dition are therefore spelled with two syllables. 

The reading consists in naming only the word Reading: in 

n i • j m, what it con- 

which expresses the final idea. 1 hus, sisl3 . 

0123456789 Examples. 

1111111111 

One two three four five six seven eight nine ten. 

We may now read the words which express 
tfie first hundred combinations. 



READINGS. Read. 

123456789 10 Two, three, 
1111111111 fOUr ' &C * 



Three, four, 
&c. 



Four, five, 
&c. 



Five, six, &a 



Six, seven, 
&c. 



Seven, eight, 
&c. 



Eight, nine, 



Nino, ten, &c 



1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


4 


4 


4 


4 


4 


4 


4 


4 


4 


4 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


5 


5 


5 


5 


5 


5 


5 


5 


5 


5 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


6 


6 


6 


6 


6 


6 


6 


6 


6 


6 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


7 


7 


7 


7 


7 


7 


7 


7 


7 


7 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 



124 



MATHEMATICAL 8CILXCE. [BOOK II. 



Ten, eleven, 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


&c. 


9 


9 


9 


9 


9 


9 


9 


9 


9 


9 


Eleven, 


l 


2 


3 


4 


5 


6 


7 


8 


9 


10 


twelve, &c. 


10 


10 


10 


10 


10 


10 


10 


10 


10 


10 



Example for § 117. In this example, beginning 

reading in . 

Addition, at the right hand, we say, 8, 17, 18, 
26 : setting down the 6 and carry- 
ing the 2, we say, 8, 13, 20, 22, 29 : 
Setting down the 9 and carrying 
the 2, we say, 9, 12, 18, 22, 30: 
and Betting down the 30, we have the entire sum 
Ail examples 3096. All the examples in addition may be done 

so solved. 

in a similar manner. 



878 
421 
679 
854 

761 
3096 



Advantages 
of reading. 



§ 118. The advantages of this method of read- 
ing over spelling are very great, 
lit. stated. 1st. The mind acquires ideas more readily 
through the eye than through either of the other 
senses. Hence, if the mind be taught to appre- 
hend the result of a combination, by merely see- 
ing its elements, the process of arriving at it is 
much shorter than when those elements are pre- 
sented through the instrumentality of sound. 
Thus, to see 4 and 4, and think 8, is a very dif- 
ferent thing from saying, four and four are eight 

2d. The mind operates with greater rapidity 
and certainty, the nearer it is brought to the 



3d. stated. 






CHAP. II.] ARITHMETIC WORDS. 125 

ideas which it is to apprehend and combine. 
Therefore, all unnecessary words load it and 
impede its operations. Hence, to spell when 
we can read, is to fill the mind with words 
and sounds, instead of ideas. 

3d. All the operations of arithmetic, beyond 3d. staled, 
the elementary combinations, are performed on 
paper ; and if rapidly and accurately done, must 
be done through the eye and by reading. Hence 
the great importance of beginning early with a 
method which must be acquired before any con- 
siderable skill can be attained in the use of 
figures. 

§ 119. It must not be supposed that the read- Reading 

ing can be accomplished until the spelling has spelling. 
first been learned. 

In our common language, we first learn the same asm 

. our common 

alphabet, then we pronounce each letter in a language, 
word, and finally, we pronounce the w r ord. We 
should do the same in the arithmetical reading. 



WORDS — SPELLING AND READING IN SUBTRACTION. 

§ 120. The processes of spelling and reading same prmci 
which we have explained in the addition of insubtrac- 
numbers, may, with slight modifications, be ap- 
plied in subtraction. Thus, if we are to subtract 



12G MATHEMATICAL SCIENCE. [BOOK II. 

2 from 5, we say, ordinarily, 2 from 5 leaves 3 ; 
or 2 from 5 three remains. Now, the word, 
three, is suggested by the relation in which 2 
and 5 stand to each other, and this word may be 
Readings in read at once. Hence, the reading, in subtrac- 

Subtraction . 7 . - 77-7 

explained. tion, ls simply naming the word, which expresses 
the difference between the subtrahend and min- 
uend. Thus, we may read each word of the 
following one hundred combinations. 



READINGS. 



One from 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


one, &c. 


1 


1 


1 


1 


1 


1 


1 


1 


1 


1 


Two from 


O 


3 


4 


5 


G 


7 


8 


9 


10 


11 


two, &c 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


Three from 


3 


4 


5 


G 


7 


8 


9 


10 


11 


12 


throe, &c. 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


Four from 


4 


5 


G 


7 


8 


9 


10 


11 


19 


13 


four, Sec. 


4 


4 


4 


4 


4 


4 


4 


4 


4 


4 


Five from 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


five, &c. 


5 


5 


5 


5 


5 


5 


5 


5 


5 


5 


Six from six, 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


&c 


G 


6 


6 


6 


G 


6 


6 


6 


6 


6 


Seven from 


7 


8 


9 


10 


11 


12 


13 


14 


15 


1G 


seven, &c. 


7 


7 


7 


7 


7 


7 


7 


7 


7 


7 



CHAP. II.] ARITHMETIC SPELLING. 127 



8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


Eight from 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 


eight, &c. 


9 


10 


11 


12 


13 


14 


15 


16 


17 


18 


Nine from 


9 


9 


9 


9 


9 


9 


9 


9 


9 


9 


nine, &c. 


10 


11 


12 


13 


14 


15 


16 


17 


18 


19 


Ten from ten, 


10 


10 


10 


10 


10 


10 


10 


10 


10 


10 


&c 



§ 121. It should be remarked, that in subtrac- 
tion, as well as in addition, the spelling of the spelling P r©- 

, .. . cedes reading 

words must necessarily precede their reading. m s U btrac- 
The spelling consists in naming the figures with tlon * 
which the operation is performed, the steps of 
the operation, and the final result. The reading Reading, 
consists in naming the final result only. 



SPELLING AND READING IN MULTIPLICATION. 

§ 122. Spelling in multiplication is similar to spelling in 

-i t , .. . , Multiplica- 

the corresponding process in addition or subtrac- t iou. 
tion. It is simply naming the two elements 
which produce the product ; whilst the reading Reading, 
consists in naming only the word which ex- 
presses the final result. 

In multiplying each number from 1 to 10 by Example** 
2, we usually say, two times 1 are 2 ; two times speilmg ' 
2 are 4 : two times 3 are 6 ; two times 4 are 8 ; 
two times 5 are 10; two times 6 are 12; two 



128 B1ATHEMATICA L SCIENCE. [BOOK II. 

times 7 are 14; two times 8 are 16; two times 
Ib reading. 9 are 18; two times 10 are 20. Whereas, we 
should merely read, and say, 2, 4, 6, 8, 10, 12, 
14, 16, 18, 20. 

In a similar manner we read the entire mul- 
tiplication table. 

I I AD I N G S. 
Once one Is 12 1 1 10 9 8 7 6 5 4 3 2 1 



Two times 1 


12 


11 


10 


9 


8 


7 


G 


."> 


4 


a 


2 


1 


are2,&a 
























2 


Tli ice times 1 


12 


11 


10 


9 


8 


7 


G 


S 


4 


3 


Q 


1 


are 3, &lc. 
























3 


Four times 1 


12 


11 


10 


9 


8 


7 


G 


5 


4 


a 


2 


1 


are 4, Sec. 
























4 


Five times 1 


12 


11 


10 


9 


8 


7 


G 


5 


4 


3 


2 


1 


are 5, &c 
























5 


Six times 1 


12 


11 


10 


9 


8 


7 


G 


5 


4 


3 


2 


1 


are six, &c. 
























6 


Seven times 


12 


11 


10 


9 


8 


7 


6 


5 


4 


3 


2 


1 


1 are 7, &c 
























7 


Eight times 1 


12 


11 


10 


9 


8 


7 


G 


5 


4 


3 


2 


1 


are 8, &c 
























8 



CnAr. II.] ARITHMETIC READING. 129 

12 11 10 9 8 7 6 5 4 3 2 1 in* times l 

q are 9, &e. 

12 11 10 9 8 7 6 5 4 3 2 1 Ten times I 

in are 10, &c. 

12 11 10 98765432 1 Eleven times 

11 1 are 11, <kc 

12 11 10 9 8 7 6 5 4 3 2 1 Twelretime. 

12 1 are 12, &c. 



SPELLING AND READING IN DIVISION. 

§ 123. In all the cases of short division, the in short vtr* 
quotient may be read immediately without nam- ^^j 
ing the process by which it is obtained. Thus, 
in dividing the following numbers by 2, we 
merely read the words below. 

2)4 6 8 10 12 16 18 22 

two three four five six eight nine eleven. 

In a similar manner, all the words, expressing i n aU ca^ 
the results in short division, may be read. 

READINGS. 

2)2 4 6 8 10 12 14 16 18 20 22 24 Twoins, 

once, &c. 

3)3 6 9 12 15 18 21 24 27 30 33 36 Three ms, 

once, &c. 

4)4 8 12 16 20 24 28 32 36 40 44 48 Four in 4, 

once, &.c 

9 



130 MATHEMATICAL SCIENCE. [BOOK II. 



Five in 5, 5)5 10 15 20 25 30 35 40 45 50 55 60 



once, &c. 




Six in 6, 


6)6 12 18 24 30 36 42 48 54 60 66 72 


once, &c. 




Seven in 7, 


7)7 14 21 28 35 42 49 56 63 70 77 84 


once, &.c. 




Eight in 8, 


8)8 16 24 32 40 48 56 64 72 80 88 96 


once, ice. 




Nine in 9, 


9) 9 18 27 36 45 54 63 72 81 90 99 108 


once, &c 




Ten in 10, 


10)10 20 30 40 50 60 70 80 90 100 110 120 


ooee, fee. 




Eleven in 1 1. 


11)11 22 33 44 55 66 77 88 99 110 121 132 


once, &.C. 




Twelve in 12, 


12)12 24 36 48 60 72 84 96 108 120 132 111 



once fee, 



UNITS INCREASING BY THE SCALE OF T 



rhe tteaofa § 124. The idea of a particular number is ne- 
nmntuTis cessarily complex ; for, the mind naturally asks: 

complex. j gt What j s t)ie unit Qr bas j s Q £ the num b er ? 



and, 

2d. How many times is the unit or basis 
taken ? 



What a fig- § 125. A figure indicates how many times a 

Uie indicates. . . _, . _ . r . 

unit is taken, li-ach ot the ten figures, however 
written, or however placed, alw T ays expresses as 
many units as its name imports, and no more ; 
nor does the figure itself at all indicate the kind 



CHAP. II.] ARITHMETIC SCALE OF TEN3. 131 



of unit. Still, every number expressed by one or Number has 

r ' 1 11 one f° r "" 

more figures, has for its basis either the abstract basi9 . 
unit one, or a denominate unit.* If a denomi- 
nate unit, its value or kind is pointed out either 
oy our common language, or as we shall present- 
ly see, by the place where the figure is written. 

The number of units which may be expressed 
by either of the ten figures, is indicated by the Number ex- 
name of the figure. If the figure stands alone, 6ingle flgure . 
and the unit is not denominated, the basis of the 
number is the abstract unit 1. 

§ 126. If we write on the right of j 

1 0, How ten is 

1, We have J written. 

which is read one ten. Here 1 still expresses 
one, but it is one ten ; that is, a unit ten times 
as great as the unit 1 ; and this is called a unit unit of the 

Of the Second Order. second order. 

Again : if we write two 0's on the ) 

t 100 How to write 

right of 1, we have J one hundred. 

which is read one hundred. Here again, 1 still 
expresses one, but it is one hundred ; that is, a 
unit ten times as great as the unit one ten, and a unit of the 
a hundred times as great as the unit 1. 



third order. 



£> L 



§ 127. If three Ts are written by ) Laws-when 

J l JJ] figures are 

the side of each other, thus - - - - ) written by 



the side of 
each other. 



* Section 111. 



132 



MATHEMATICAL SCIENCE. [BOOK II. 



First. 



Second. 



Third. 



the ideas, expressed in our common anguage, 
are these : 

1st. That the 1 on the right, will either express 
a single thing denominated, or the abstract unit 
one. 

2d. That the 1 next to the left expresses 1 ten 
that is, a unit ten times as great as the first. 

3d. That the 1 still further to the left expresses 
1 hundred; that is, a unit ten times as great as 
the second, and one hundred times as gtcai OS the 
first; and similarly if titer fhrr jila- 

When figures are thus written by the side of 
each other, the arithmetical language establishes 
when figures ft re ] al j 01l between the units of their places : that 

are so writ- ■ 

ton - is, the unit of each place, ns we pass from the 
right hand towards the left, increases according 
to the scale of tens. Therefore, by a law of the 

arithmetical language, the place of a figure Ji 
its unit. 

If, then, we write a row of 0\s as a scale, 
thus : 



What the 
language 

establishes 



Scale for 
Numeration. 



f 

lit 

I li 

r& t-> rO 



S c 3 



B 

1 

■ 

II- 

O O § O 

J g J 111 

i^3 -t-> -t-> *** -*j 3 



Theunitsof 00 0, 00 0, 00 0, 000 

place deter- 
mined, the unit of each place is determined, as well 



CHAP. II.] ARITHMETIC SCALE OF TENS. 133 



as the law of change in passing from one place 

to another. If then, it were required to express How any 

~ ~ number of 

a given number of units, of any order, we first units may be 
select from the arithmetical alphabet the char- expre! 
acter which designates the number, and then 
write it in the place corresponding to the order. 
Thus, to express three millions, we write 

3000000 ; 
and similarly for all numbers. 

§ 128. It should be observed, that a figure a figure has 

7 7 • 7 7 no value in 

oeing a character which represents value, can iUself> 
have no value in and of itself. The number of 
things, which any figure expresses, is determined 
by its name, as given in the arithmetical alpha- 
bet. The kind of thing, or unit of the figure, is iiowtheunu 
fixed either by naming it, as in the case of a de- BdmsAt 
nominate number, or by the place which the 
figure occupies, when written by the side of or 
over* other figures. 

The phrase "local value of a figure," so long Figure, has 
in use, is, therefore, without signification when n ^ e< 
applied to a figure : the term * local value/' 
being applicable to the unit of the place, and Terma PP ii 
not to the figure which occupies the place. unf^jL 

§ 129. Federal Money affords an example of a Federal 

"' Money 

* Section 199. 



134 MATHEMATICAL SCIENCE [BOOK II. 

its denomina- series of denominate units, increasing according 
to the scale of tens : thus, 

*> a a f d 
w P P u S 
11111 

How read, may be read 11 thousand 1 hundred and 11 

mills; or, 1111 cents and 1 mill; or, 111 dimes 

] cent and 1 mill; or, 11 dollars 1 dime 1 cent 

and 1 mill; or, 1 eagle 1 dollar 1 dime 1 cent 

variou*kiP43 and 1 mill. Thus, we may read the number 

of Readings. ^.^ eitner Q f j ts m ^ § as a busis, or we may 

name them all : thus, 1 eagle, 1 dollar, 1 dime, 
1 cent, 1 mill. Generally, in Federal Money, 
we read in the denominations of dollars, cents, 
and mills; and should say, 11 dollars 11 cents 
and 1 mill. 



Fxnmpiesin § 130. Examples in reading figures :— 
inEompta. If we have the figures - - - - 89 

we may read them by their smallest 

unit, and say eighty-nine; or, we may say 8 

tens and 9 units. 
2d. Example. Again, the figures 567 

may be read by the smallest unit; 

viz. five hundred and sixty-seven; or we may 

say, 56 tens and 7 units ; or, 5 hundreds 6 tens 

and 7 units. 
u. E^upie. Again, the number expressed by - 74896 



CHAP. II.] ARITHMETIC VARYING SCALES. 135 

may be read, seventy-four thousand eight hun- various read- 

insrs of a 

dred and ninety-six. Or, it may be read, 7489 number. 
tens and 6 units ; or, 748 hundreds 9 tens and 
6 units : or, 74 thousands 8 hundreds 9 tens 
and 6 units ; or, 7 ten thousands 4 thousands 8 
hundreds 9 tens and 6 units : and we may read 
in a similar way all other numbers. 

Although we should teach all the correct read- The best 
ings of a number, we should not fail to remark rea ding. 
that it is generally most convenient in practice 
to read bv the lowest unit of a number. Thus, 
in the numeration table, we read each period by Each period 
the lowest unit of that period. For example, in ^westunu. 
the number 

874,967,847,047, Example. 

we read 874 billions 967 millions 847 thousands 
and 47. 



UNITS INCREASING ACCORDING TO VARYING SCALES. 

§ 131. If we write the well-known signs of Methods of 
the English money, and place 1 under each de- ^es having 

different 

denominate 

units. 



nomination, we shall have 

£. s. cL /. 
1111 

Now, the signs £.s. d. and/, fix the value of How the 
the unit 1 in each denomination; and they also unit is fixed, 



136 MATHEMATICAL SCIENCE. [BOOK H. 

what the determine the relations which subsist between 
expresses, the different units. For example, this simple 

language expresses these ideas : 
The units of 1st. That the unit of the right-hand place is 
1 farthing — of the place next to the left, 1 penny 
— of the next place, 1 shilling — of the next place, 
1 pound ; and 
How the 2d. That 4 units of the lowest denomination 
increase, make one lirnt of tne next higher; 12 of the 
second, one of the third; and 20 of the third, 
one of the fourth. 
TheunitMin If we take the deiioi ninate numluM's of the 
Avoirdupois weight, we have 



weight. 



7'"n. CWL yr. lb. oz. dr. 

111111; 
chants in ID which the units increase in the following 
flwimitai mann © r: vv/ -- tn0 wcond unit, counting from 

the right, is sixteen times as great as the G 

the third, sixteen times as great Bfl the second; 
the fourth, twenty-five times as great as the 
third ; the fifth, four times as great as the fourth ; 
and the sixth, twenty times as great as the fifth. 
How the scale The scale, therefore, for this class of denominate 

numbers varies according to the above laws. 

a different If we take any other class of denominate 

, J , I<HI numbers, as the Troy weight, or any of the 

systems of measures, we shall have different 

scales for the formation of the different units. 



CIMP.II.] ARITHMETIC INTEGER UNITS. 137 



But in all the formations, we shall recognise The method 
the application of the same general principles. the scales the 
There are, therefore, two general methods of Ambers* 11 
forming the different systems of integer num- 

Two systems 

bers from the unit one. The first consists in of forming 
preserving a constant law of relation between lne ^ mi 
the different unities ; viz. that their values shall 

First system, 

chancre according to the scale of tens. This 
gives the system of common numbers. 

The second method consists in the application second sys- 
of known, though varying laws of change in the 
unities. These changes in the unities produce change in the 
the entire system of denominate numbers, each for ^ingthe 
class of which has its appropriate scale, and the uml,es - 
changes among the units of the same class are 
indicated by the different degrees of its scale. 



INTEGER UNITS OF ARITHMETIC. 

§ 132. There are four principal classes of units Four classes 
in arithmetic : 

1st. Abstract, or simple units; 1st. class. 

2d. Units of Currency ; ^^ 

3d. Units of Weight ; and ^ c]ass> 

4th. Units of Measure. 4th. class. 

First among the Units of arithmetic stands 
the simple or abstract unit 1. This is the basis Abstract unit 
of all simple numbers., and becomes the basis,. one ' thebasi8 - 



138 MATHEMATICAL SCIENCE. [BOOK II. 



The basis of also, of all denominate numbers, by merely na- 

denominate . . . 

numbers; mmg, in succession, the particular things to 

which it is applied. 
Also, the ba- It is also the basis of all fractions. Merely as 

dis of all frac- 
tions, the unit 1, it is a whole which may be divided 

whether eim- , . i r • • . r 

pieordenom- according to any law, forming every variety oi 

mate * fraction ; and if we apply it to a particular thing, 

the fraction becomes denominate, and we have 

expressions for all conceivable parts of that thing. 



§ 133. It has been remarked 1 that we can 

Must nppn- f orm , 1() distinct ; , i ,j , 1V 1 iei isi< m of a number, un- 
bead the 

unit. til we have a elixir notion of its unit, and the 

number of times the unit is taken. The unit is 

the great basis. The utmost care, therefi 

Lot it* nature should be taken to impress on the minds of 

and kind be .... . . . . 

fully explain- learners, a clear and distinct idea 01 t he actual 
value of the unit of every number with which 
they have to do. If it be a number expressing 
currency, one or more of the coins should be 

preying cur- exhibited, and the value dwelt upon; after which, 

rency. 

distinct notions of the other units can be ac- 
quired by comparison. 
_ . . , If the number be one of weight some unit 

Exhibit the ° 

unit if it be should be exhibited, as one pound, or one ounce, 

of weight ; 

and an idea of its weight acquired by actually 



ed; 



J low for a 
number ex- 



* Section 110. 



CHAP. II.] ARITHMETIC FEDERAL MONEY. 130 

lifting it. This is the only way in which we 
can learn the true signification of the terms. 

If the number be one of measure, either And aiso,ifu 
linear, superficial, liquid, or solid, its unit should measure! 
also be exhibited, and the signification of the 
term expressing it, learned in the only way in 
which it can be learned, through the senses, and 
by the aid of a sensible object. 

FEDERAL MONEY. 

§ 134. The currency of the United States is currency of 
called Federal Money. Its units are all denomi- theL - States - 
nate, being 1 mill, 1 cent, 1 dime, 1 dollar, 1 
eagle. The law of change, in passing from one Law of 
unit to another, is according to the scale of tens, unties. 
Hence, this system of numbers may be treated, 

How these 

in all respects, as simple numbers; and indeed numbers may 

be treated. 

they are such, with the single exception that 
their units have different names. 

They are generally read in the units of dollars, How gen- 
cents, and mills — a comma being placed after 
the figure denoting dollars. Thus, 

#864,849 ExampIe> 

is read eight hundred and sixty-four dollars, 
eighty-four cents, and nine mills; and if there 
were a figure after the 9, it would be read in 0ffi ^ re9 

after milieu 

decimals of the mill. The number mav, how- 



140 MATHEMATICAL SCIENCE. [BOOK II. 

The number ever, be read in any other unit ; as, 8G4849 

read in 

various ways, mills ; or, 86484 cents and 9 mills; or, 8648 
dimes, 4 cents, and 9 mills ; or, 86 eagles, 4 dol- 
lars, 84 cents, and 9 mills; and there are yet 
several other readings. 

ENGLISH MONEY. 

sterling Mo- § 135. The units of English, or Sterling Mo- 
1C> * ney, are 1 farthing, 1 penny, 1 shilling, and 1 
pound. 

scaieofthe The scale of this class of numbers is a varying 
scale. Its degrees, in passing from the unit of 
the lowest denomination to the highest, arc four, 

How it twelve, and twenty. For, four farthings make 
one penny, twelve pence one shilling, and twenty 
shillings one pound. 



unities. 



changes. 



AVOIRDUPOIS WEIGHT. 

Units in § 13G. The units of the Avoirdupois Weight 
are 1 dram, 1 ounce, 1 pound, 1 quarter, 1 hun- 
dred-weight, and 1 ton. 
scale. The scale of this class of numbers is a vary- 

ing scale. Its degrees, in passing from the unit 
of the lowest denomination to the highest, are 
sixteen, sixteen, twenty-five, four, and twenty, 
v-iriationin ^ 0T > s i xteen drams make one ounce, sixteen 
in degrees, ounces one pound, twenty-five pounds one quar 



CHAP. II. J ARITHMETIC UNITS OF LENGTH. 141 

ter, four quarters one hundred, and twenty hun- 
dreds one ton. 

TROY WEIGHT. 

§ 137. The units of the Troy Weight are, 1 Units in 

J Troy Weight. 

grain, 1 pennyweight, 1 ounce, and 1 pound. 

The scale is a varying scale, and its degrees, scale: 
in passing from the unit of the lowest denomina- Jts Agrees. 
tion to the highest, are twenty-four, twenty, and 
twelve. 

APOTHECARIES' WEIGHT. 

§ 138. The units of this weight are, 1 grain, 1 units in 

i -, j , , _ , Apothecaries 

scruple, 1 dram, 1 ounce, and 1 pound. weight 

The scale is a varying scale. Its degrees, in s^e: 
passing from the unit of the lowest denomina- its degrees, 
tion to the highest, are twenty, three, eight, and 
twelve. 

UNITS OF MEASURE. 

§ 139. There are three units of measure, each T ^eeuniu 

of measure. 

differing in kind from the other two. They are, 
Units of Length, Units of Surface, and Units of 
Solidity. 

UNITS OF LENGTH. 

§ 140. The unit of length is used for measur- LTnits of 



ing lines, either straight or curved. It is a 



length. 



142 



MATHEMATICAL SCIENCE. 



[book II. 



The stand- straight line of a given length, and is often called 

ard. 

the standard of the measurement. 



What units 
are taken. 



Idea of 

length. 



The units of length, generally used as stand- 
ards, are 1 inch, 1 foot, 1 yard, 1 rod, 1 furlong, 
and 1 mile. The number of times which the 
unit, used as a standard, is taken, considered in 
connection with its value, gives the idea of the 
length of the line measured. 



UNITS OF SURFACE. 



Units of 
surface. 



What the 

unit of 

surface is. 



Examples. 



Its connection 

with the unit 

of length. 



Square feet 

in a 
gquare yard. 



l MjtMura foot 



§ 141. Units of surface are used for the meas- 
urement of the area or contents of whatever has 
the two dimensions of length and breadth.- The 

unit of surface is a square de- 
scribed on the unit of length 
as a side. Thus, if the unit 
of length be 1 foot, the corre- 
sponding unit of surface will 
be 1 square foot; that is, a square constructed on 
1 foot of length as a side. 

If the linear unit be 1 yard, 
the corresponding unit of sur- 
face will be 1 square yard. It 
will be seen from the figure, 
that, although the linear yard 
contains the linear foot but 
three times, the square yard 



1 vard. 






i 



CHAP. II. J ARITHMETIC DUODECIMAL UMTS. 143 



contains the square foot nine times. The square square rod 
rod or square mile may also be used as the unit square miic. 
of surface. 

The number of times which a surface contains Area or 

contents of a 



surface. 



its unit of measure, is its area or contents ; and 
this number, taken in connection with the value 
of the unit, gives the idea of its extent. 

Besides the units of surface already considered, 
there is another kind, called, 



DUODECIMAL UNITS. 

§ 142. The duodecimal units are generally Duodecimal 
used in board measure, though they may be used units# 
in all superficial measurements, and also in solid. 

The square foot is the basis of this class of Their basis, 
units, and the others are deduced from it, by a 
descending scale of twelve. 

§ 143. It is proved in Geometry, that if the What prmci . 
number of linear units in the base of a rectan- ple * s proved 

in Geometry. 

gle be multiplied by the number of linear units 
in the height, the numerical value of the pro- 
duct will be equal to the number of superficial 
units in the figure. 

Knowing this fact, we often express it by say- H owitia ex- 
ing, that "feet multiplied by feet give square pressed * 
feet/' and "yards multiplied by yards give square 



144 MATHEMATICAL SCIENCE. [BOOK II. 

This a concise yards." But as feet cannot be taken fe^t times, 
expre. ion. ^^ y ar( j s yard times, this language, rightly un- 
derstood, is but a concise form of expression for 
the principle stated above, 
conclusion. With this understanding of the language, we 
say, that 1 foot in length multiplied by 1 foot in 
height, gives a square foot ; and 4 feet in length 
multiplied by 3 feet in height, gives IS squaro 
feet. 



Examples in § 144. If DOW, 1 foot 10 

the rnultipli- t _ . . 

cation of feet length be multiplied by 1 men 

b taSL. =t-2 °f a foot in height, the 
product will be one-twelfth 

of a square foot ; that is. one- 
twelfth of the first unit : if it 
be multiplied by 3 inches, the product will be 
Generaiiza- three-twelfths of a square fool ; and similarly 

tion. ' 

for a multiplier of any number of inch 
inches by If, now, we multiply 1 inch by 1 inch, the 
product may be represented by 1 square inch : 
How the units that is, by one-twelfth of the last unit. Hence, 

change, and , . . „ 

what they the units of this measure decrease according to 
the scale of 12. The units are, 
First. 1st. Square feet — arising from multiplying feet 

by feet. 

second. 2d. Twelfths of square feet — arising from mul- 

tiplying feet by inches. 






CHAP. II.] ARITHMETIC UNITS. 145 

3d. Twelfths of twelfths — arising from multi- Third, 
plying inches by inches. 

The same remarks apply to the smaller di- conclusion 
visions of the foot, according to the scale of 
twelve. 

The difficulty of computing in this measure Difficulty, 
arises from the changes in the units. 



UNITS OF SOLIDITY. 

§ 145. It has already been stated, that if units of soii- 
length be multiplied by breadth, the product 
may be represented by units of surface. It is what is 
also proved, in Geometry, that if the length, ometry iu re _ 
breadth, and height of any regular solid body, gardtolnera - 
of a square form, be multiplied together, the 
product may be represented by solid units whose 
number is equal to this product. Each solid solid units, 
unit is a cube constructed on the linear unit as 
an edge. Thus, if the linear unit be 1 foot, the Examples, 
solid unit will be 1 cubic or solid foot ; that is, 
a cube constructed on 1 foot as an edge ; and 
if it be 1 yard, the unit will be 1 solid yard. 

The three units, viz. the unit of length, the The three 
unit of surface, and the unit of solidity, are es- tiaiiydiffer. 
sentially different in kind. The first is a line ent * 
of a known length ; the second, a square of a wiw* they 
known side : and the third, a solid, called a aw * 

10 



146 MATHEMATICAL SCIENCE. [BOOK II. 



Generally cube, of a known base and height. These are 
the units used in all kinds of measurement — ■ 

Duodecimal excepting only the duodecimal system, which 
has already been explained. 



LIQUID MEASURE. 

Units of Li- § 146. The units of Liquid Measure are, 1 

quid Meas- 

ure. gill, 1 pint, 1 quart, 1 gallon, 1 barrel, 1 hogs- 



head, 1 pipe, 1 tun. The scale is a varying 
scale. Its degrees, in passing from the unit of 
iiowitva- the lowest denomination, are, four, two, four, 
thirty-one and a half, sixty-three, two, and two. 



Scale. 



ries. 



DRY MEASURE. 

Units of Dry § 147. The units of this measure are, 1 pint, 

Measure. 

1 quart, 1 peck, 1 bushel, and 1 chaldron. The 

Degrees of degrees of the scale, in passing from units of the 

lowest denomination, are two, eight, four, and 



thirty-six. 



TIME. 



Unitsof § 148. The units of Time are, 1 second, 1 

minute, 1 hour, 1 day, 1 week, 1 month, 1 year, 
Degrees of and 1 century. The degrees of the scale, in 
passing from units of the lowest denomination to 
the highest, are sixty, sixty, twenty-four, seven, 
four, twelve, and one hundred. 



CHAP. II.] ARITHMETIC ADVANTAGES. 147 



CIRCULAR MEASURE. 



§ 149. The units of this measure are, a sec- umtsofc» 

cular Meaa 

ond, 1 minute, 1 degree, 1 sign, 1 circle. The ure. 

degrees of the scale, in passing from units of the Degrees ot 

& . the Scale. 

lowest denomination to those of the higher, are 
sixty, sixty, thirty, and twelve. 



ADVANTAGES OF THE SYSTEM OF UNITIES. 

§ 150. It may well be asked, if the method Ad'an.agoa 

of the sa stem 

here adopted, of presenting the elementary prin- 
ciples of arithmetic, has any advantages over 
those now in general use. It is supposed to pos- 
sess the following : 

1st. The svstem of unities teaches an exact 1st Teaches 

%> 

analysis of all numbers, and unfolds to the mind f numbers: 
the different ways in which they are formed from 
the unit one, as a basis. 

2d. Such an analysis enables the mind to form 2d.p intsout 
a definite and distinct idea of every number, by relation: 
pointing out the relation between it and the unit 
from which it was derived. 

3d. By presenting constantly to the mind the 3d. constant- 
ldea of the unit one, as the basis of all numbers, the id g a of 
the mind is insensibly led to compare this unit xa ^ tJu 
with all the numbers which flow from it and 



148 



MATHEMATICAL SCIENCE. 



[book II. 



then it can the more easily compare these num- 
bers with each other. 
4th. Explains 4th. It affords a more satisfactory analysis, 
the four an d a better understanding of the four ground 
ground Ylx ] eSj an d indeed of all the operations of arith- 

rules. x 

metic, than any other method of presenting the 



subject. 



FOUR GROUND RULES. 



system § 151. Let us take the two following examples 

"addition. m Addition, the one in simple and the other in 
denominate numbers, and then analyze the pro- 
cess of finding the sum in each. 



Examples. 



slMIM.i: SUH 

874198 

36984 

3641 



911823 



M'.VinEM 

cwt. (jr. lb. oz. dr. 

3 3 24 15 14 

G 3 23 14 8 

10 3 23 14 6 



Process of 

performing 

addition. 



In both examples we begin by adding the units 
of the lowest denomination, and then, we divide 
their sum by so many as make one of the denomi- 
nation next higher. We then set down the 
remainder, and add the quotient to the units 
of that denomination. Having done this, we 
apply a similar process to all the other denomina- 
Butone tions — the principle being precisely the same in 
both examples. We see, in these examples, an 



CHAP. II.] ARITHMETIC SUBTRACTION. 



149 



illustration of a general principle of addition, Units of the 

same kind 

viz. that units of the same kind are always added ^to. 
together. 



§ 152. Let us take two similar examples in system 

applied in 
bUDtractlOn. subtraction. 



SIMPLE NUMBERS. 


DENOMINATE NUM 


BERS. 


8403 
3298 


£ 5. cL 

6 9 7 


far. 
2 


5105 


3 10 8 


4 




2 18 10 


2 



Examples. 



In both examples we begin with the units of The method 

[ of performing 

the lowest denomination, and as the number in the examples, 
the subtrahend is greater than in the place di- 
rectly above, we suppose so many to be added 
in the minuend as make one unit of the next 
higher denomination. We then make the sub- 
traction, and add 1 to the units of the subtrahend 
next higher, and proceed in a similar manner, 
through all the denominations. It is plain that 
the principle employed is the same in both exam- Principle the 
pies. Also, that units of any denomination in a u examples 
the subtrahend are taken from those of the same 
denomination in the minuend. 



§ 153. Let us now take similar examples in Multiplies. 

«,,.,. . tion. 

Multiplication. 



150 



MATHEMATICAL SCIENCE. [BOOK II. 



Examples. 



SIMPLE NUMBERS. 

87464 
5 



437320 



DENOMINATE NUMBERS. 

ft I 3 3 gr. 

9 7 6 2 15 

5 

48 3 2 1 15 



Mtthod of In these examples we see, that we multiply, in 

perf °| ming succession, each order of units in the multipli- 

exampies. can <i by the multiplier, and that we carry from 

one product to another, one for every so many as 

make one unit of the next higher denomination. 

The princi- ° 

pie the same The principle of the process is therefore the 

for all 

examples, same in both examples. 

§ 154. Finally, let us take two similar exam- 
Division. p]es in Divisjolh 



Examples. 



simti.i: KUKBXB& 


DENO 


MINAT 




IBKRS. 


3)874911 


£ 


S. 


d. 


far. 


291637 


3)8 


4 


2 


1 




2 


14 


8 


3 



Principles in- We begin, m both examples, by dividing the 
voived: un j( s f the highest denomination. The unit of 
the quotient figure is the same as that of the 
dividend. We write this figure in its place, and 
then reduce the remainder to units of the next 
lower denomination. We then add in that de- 

The same as . . . . . 

in the nomination, and continue the division through 
other rules. ^ ^ denominations to the last — the principle 
being precisely the same in both examples. 



CHAP. II.] ARITHMETIC FRACTIC VS. 151 



SECTION II. 



FRACTIONAL UNITS. 



FRACTIONAL UNITS. SCALE OF TENS. 

§ 155. If the unit 1 be divided into ten equal Fraction on©, 
parts, each part is called one tenth. If one of de ^ ied . 
these tenths be divided into ten equal parts, 
each part is called one hundredth. If one of the hundredth ; 
hundredths be divided into ten equal parts, each 0ne 
part is called one thousandth ; and corresponding thousandth ' 
names are given to similar parts, how far soever Generaiiza- 
the divisions may be carried. 

Now, although the tenths which arise from Fractions are 
dividing the unit 1, are but equal parts of 1, t T hole 
they are, nevertheless, whole tenths, and in this 
light may be regarded as units. 

To avoid confusion, in the use of terms, we Fractional 
shall call every equal part of 1 a fractional unit 
Hence, tenths, hundredths, thousandths, tenths 
of thousandths, &c, are fractional units, each 
having a fixed relation to the unit 1, from which 
it v»as derived. 



152 MATHEMATICAL SCIENCE. [BOOK IT. 



Fractional § 156. Adopting a similar language to that 

units of the 

first order; used in integer numbers, we call the tenths, frac- 
deT&c?" ti° na l un i ts °f the first order ; the hundredths, 
fractional units of the second order ; the thou- 
sandths, fractional units of the third order ; and 
so on for the subsequent divisions. 
Language for * s there any arithmetical language by which 
fractional t ] 1(>>( > fractional units may be expressed? The 

units. J ■ 

decimal point, which is merely a dot, or period, 
Whatitfix^. indicates the division of the unit 1, according to 

the scale of t&ns. By the arithmetical language, 
Nnmcsofthe the unit of the place next the point, on the right, 

places, 

is 1 tenth ; that of the second place, 1 hun- 
dredth ; that of the third, 1 thousandth; that of 
the fourth, 1 ten thousandth; and so on for 
places still to the right. 
s<»ie. The scale for decimals, therefore. 

.000000000. etc. ; 

in whicii the unit of each place is known as 
soon as we have learned the signification of the 
language. 

If, therefore, we wish to express any of the 
parts into which the unit 1 may be divided, ac- 
cording; to the scale of tens, we have simply to 

Any decimal ° r J 

number may se lect from the alphabet, the figure that will 

be expressed 

bythisscaie. express the number of parts, and then write it in 



CHAP. II.] ARITHMETIC FRACTIONS. 153 

the place corresponding to the order of the unit, where any 

figure is 

Thus, to express four tenths, three thousandths, written. 
eight ten-thousandths, and six millionths, we 
write 

.403806 ; Exarnple. 

and similarly, for any decimal which can be 
named. 

§ 157. It should be observed that while the 
units of place decrease, according to the scale of 
tens, from left to right, they increase according Ti»»uniuin 

civ.ase from 

to the same scale, from right to left. This is the ri^nitokft. 
same law of increase as that which connects the 
units of place in simple numbers. Hence, simple consequent 
numbers and decimals being formed according to 
the same law, may be written by the side of each 
other and treated as a single number, by merely 
preserving the separating or decimal point. 
Thus, 8974 and .67046 may be written 

8974.67046 ; Example. 

since ten units, in the place of tenths, make the 
unit one in the place next to the left. 

FRACTIONAL UNITS IN GENERAL. 

§ 158. If the unit 1 be divided into two equal a halt 
parts, each part is called a half. If it be divided 



Generally. 



154 MATHEMATICAL SCIENCE. [BOOK II. 

a third, into three equal parts, each part is called a third: 

if it be divided into four equal parts, each part is 

a fourth, called a fourth: if into five equal parts, each 

a fifth, part is called a fifth ; and if into any number of 

equal parts, a name is given corresponding to the 

number of parts. 

These units Now, although these halves, thirds, fourths, 

nre whole 

things. fifths, &c, are each but parts of the unit 1, they 
are, nevertheless, in t/tcmsclvcs, whole things 
Rxampka. That IS, a half is a whole half; a third, a whole 
third ; a fourth, a whole fourth ; and the same 
for any other equal part of 1. In this sense, 
therefore, they are units, and we call them frac- 

Hasr;nvi;i- ti<>iuil units. Each is an exact pa^t of the unit 

Hon to unitv. 

1, and has a iixed relation to it. 

§ 159. Is there any arithmetical language by 
which these fractional units can be expressed? 
Lnn-u : .w tor The bar, written at the right, is the 

fractions. . . > ..... r . 

sign which denotes the division ot the 

urit 1 into any number of equal parts. 

To express If we wish to express the number of equal 

of equal parts into which it is divided, as 9, for 

parl3 ' example, we simply write the 9 under 

the bar, and then the phrase means, that some 
thing regarded as a whole, has been divided into 
9 equal parts. 









CHAP. II. J ARITHMETIC FRACTIONS. 



155 



If, now, we wish to express any 
number of these fractional units, as 7, 
for example, we place the 7 above the 
line, and read, seven ninths. 



To show how 
7 many are 

~J7" taken. 



§ 160. It was observed,* that two things are 
necessary to the clear apprehension of an inte- 
ger number. 

1st. A distinct apprehension of the unit which 
forms the basis of the number ; and, 

2dly. A distinct apprehension of the number 
of times which that unit is taken. 

Three things are necessary to the distinct ap- 
prehension of the value of any fraction, either 
decimal or vulgar. 

1st. We must know the unit, or whole thing, 
from which the fraction was derived ; 

2d. We must know into how many equal parts 
that unit is divided ; and, 

3dly. We must know how many such parts 
are taken in the expression. 

The unit from which the fraction is derived, 
is called the unit of the fraction ; and one of 
the equal parts is called, the unit of the expres- 
sion. 

For example, to apprehend the value of the 



Two things 

necessary to 

apprehend a 

number. 

First. 



Second 



Three thiogi 

necessary to 

apprehend a 

fraction. 

First 



Second. 



Third. 



Unit of the 
fraction — of 
the expres- 
sion. 



Section 110. 



156 



MATHEMATICAL SCIENCE. 



[book 



what we fraction f of a pound avoirdupois, or jib. ; we 

must know. 

must know, 



First. 
Second. 



Third. 



Unit when 
not named. 



1st. What is meant by a pound ; 

2d. That it has been divided into seven equal 
parts ; and, 

3d. That three of those parts are taken. 

In the above fraction, 1 pound is the unit 6f 
the fraction ; one-seventh of a pound, the unit 
of me expression; and 3 denotes that three fiac- 
tional units are taken. 

If the unit of a fraction be TOt named, it is 
taken to be tiie abstract unit 1. 



A D V A N T A (r E S V V B A C T I -V A L N I T S . 



Every equal § 161 . By con>i< ieri 1 1 L r every equal part ofuni- 

part of one, o 

unit. tv as a unit of itself having a certain relation to 
the unit 1, the mind is led to analyze a frac- 
tion, and thus to apprehend its precise significa- 
tion. 

Under this searching analvsis, the mind at 
once seizes on the unit of the fraction as the 
principal basis. It then looks at the value of 
each part. It then inquires how many such parts 
are taken. 
Equal units, It having been shown that equal integer units 

whether in- 
tegral or frac- can alone be added, it is readily seen that the 



Advantages 

of the 

analysis. 



CHAP. II.] ARITHMETIC ADVANTAGES. 157 

same principle is equally applicable to frac- tionai, can 

, , , , , alone be 

tionai units ; and then the inquiry is made : added> 
What is necessary in order to make such units 
equal ? 

It is seen at once, that two things are neces- Two lhin ^ 

necessary for 
Saiy : addition. 

1st. That they be parts of the same unit ; and, First - 
2d. That they be like parts ; in other words, second. 

they must be of the same denomination, and 

have a common denominator. 

In regard to Decimal Fractions, all that is Decimal 

. Fractions. 

necessary, is to observe that units of the same 
value are added to each other, and when the 
figures expressing them are written down, they 
should always be placed in the same column. 



§ 162. The great difficulty in the management Diffi cm\vin 

the manage- 

of fractions, consists in comparing them with ment of frac 

lions. 

each other, instead of constantly comparing them 
with the unity from which they are derived. 
By considering them as entire things, having a 

How 

fixed relation to the unity which is their basis, obviated, 
they can be compared as readily as integer num- 
bers ; for, the mind is never at a loss w T hen it 
apprehends the unit, the parts into which it is 
divided, and the number of parts which are JSSfc 
taken. The only reasons why we apprehend and pUcity "* 

^ * l integers. 



158 



MATHEMATICAL SCIENCE. [BOOK II. 



handle integer numbers more readily than frac- 
tions, are, 
First 1st. Because the unity forming the basis is 

always kept in view; and, 
seond. 2d. Because, in integer numbers, we have 

been taught to trace constantly the connection 
between the unity and the numbers which come 
from it ; while in the methods of treating frac- 
tions, these important considerations have bee:; 
neglected. 



[OK III. 



PROPORTION AND RATIO. 



Proportion § 1G3. Proportion expresses the relation which 

defined. ■ 

one number bears to another, with respect to its 
being greater or less, 
^'wo ways of Two numbers may be compared, the one with 

comparing. 

the other, in two ways : 
isi method. 1st. With respect to their difference, called 

Arithmetical Proportion ; and, 
2d method. 2d. With respect to their quotient, called 

Geometrical Proportion. 



CHAP. II. J ARITHMETIC PROPORTION. 259 



Thus, if we compare the numbers 1 and 8, Example of 
by their difference, we find that the second ex- portion. 
ceeds the first by 7 : hence, their difference 7, 
is the measure of their arithmetical proportion, 

J A A Arithmetical 

and is called, in the old books, their arithmetical Ratio. 
ratio. 

If we compare the same numbers by their Examp i eof 
quotient, we find that the second contains the 

Proportion. 



Ratio. 



Geometrical 

first 8 times : hence, 8 is the measure of their 
geometrical proportion, and is called their geo- 
metrical ratio* 

§ 164. The two numbers which are thus corn- 
Terms. 

pared, are called terms. The first is called the Antecedent. 
antecedent, and the second the consequent. consequent. 

In comparing numbers with respect to their comparison 
difference, the question is, how much is one ' * erence * 
greater than the other ? Their difference affords 
the true answer, and is the measure of their pro- 
portion. 

In comparing numbers with respect to their comparison 
quotient, the question is, how many times is one yqu0len 
greater or less than the other ? Their quotient 
or ratio, is the true answer, and is the measure 



* The term ratio, as now generally used, means the quo- 
tient arising from dividing one number by another. We 
shall use it only in this sense. 



160 MATHEMATICAL SCIENCE. [BOOK II. 

Example by of their proportion. Ten, for example, is 9 

difference. 

greater than 1, if we compare the numbers one 

and ten by their difference. But if we compare 

By quotient, them by their quotient, ten is said to be ten 

"Ten times." times as great — the language "ten times' 5 having 

reference to the quotient, which is always taken 

as the measure of the relative value of two 

Examples of numbers so compared. Thus, when v 

this use orthe t ^ at t | le un j ts f our common system of numbers 

term. J 

increase In a tenfold ratio, we mean that they so 

increase that each succeeding unit shall contain 

the preceding one ten times. This is a conven- 

Conveuient &M language U) express B ] >a T t icul a i* relation of 

language. ^ Q num l H . rs? nm | j s perfectly Correct, wllCll 

used iii conformity to an accurate definition. 

in what § 1 (')">• All authors agree, that the measure of 

a agreeT tne geometrical proportion, between two num- 
bers, is their ratio ; but they are by no means 
biM«» unanimous, nor does each always agree with 
grce * himself, in the manner of determining this ratio. 
Some determine it, by dividing the first term by 
Different me- the second ; others, by dividing the second term 
by the first.* All agree, that the standard, what- 

Standard the 

divhor. ever it may be, should be made the divisor. 

* The Encyclopedia Metropolitana, a work distinguished 
by the excellence of its scientific articles, adopts the latter 
method. 



CHAP. II.] ARITHMETIC RATIO. 161 

This leads us to inquire, whether the mind what is the 

r> ti i /» it best form. 

fixes most readily on the first or second number 
as a standard ; that is, whether its tendency is 
to regard the second number as arising from the 
first, or the first as arising from the second. 

§ 166. All our ideas of numbers begin at origin of 

numbers. 

one.* This is the starting-point. We con- 
ceive of a number only by measuring it with H ow we con 
one, as a standard. One is primarily in the ^^er* 
mind before we acquire an idea of any other 
number. Hence, then, the comparison begins Where the 
at one, which is the standard or unit, and all C0 ^ ari30n 

begins. 

other numbers are measured by it. When, there- 
fore, we inquire what is the relation of one to 
any other number, as eight, the idea presented The idea 
is, how many times does eight contain the stand- P re9ented . 
ard ? 

We measure by this standard, and the ratio is standard. 

Ratio. 

the result of the measurement. In this view of 

the case, the standard should be the first number Whatthe y 

should be. 

named, and the ratio, the quotient of the second 
number divided by the first. Thus, the ratio of 
2 to 6 would be expressed by 3, three being the Example, 
number of times which 6 contains 2. 



* Section 104. 
11 



162 MATHEMATICAL SCIENCE. [BOOK II. 

other reasons § 167. The reason for adopting this method 

for this me- 
thod of com- ot comparison will appear still stronger, if we 

panson. ta j^ fractional numbers- Thus, if we seek the 
relation between one and one-half, the mind im- 
mediately looks to the part which one-half is of 
comparison one , and this is determined by dividing one-half 

of unity with 

fractions, by 1 ; that is, by dividing the second by the 
first : whereas, if we adopt the other method, 
We divide our standard, and find a quotient 2. 



ceometricai § 168. It may be proper here to observe, that 

proportion. ? . 

while the term " geometrical proportion" is used 

to express the relation of two numbers, com- 

Ageometrt- pared by their ratio, the term, " a geometrical 

»*al propor- . ,, . ... r . . 

tk» defined proportion, is applied to four numbers, in which 
the ratio of the first to the second is the same as 
that of the third to the fourth. Thus, 

Example. 2 I 4 I : 6 I 12, 

is a geometrical proportion, of which the ratio 
is 2. 



Further ad- § 169. We will now state some further ad- 
vantages which result from regarding the ratio 
as the quotient of the second term divided by 
the first. 

Questions m Every question in the Rule of Three is a 

the Rule of . 

Three- geometrical proportion, excepting only, that the 



CHAP. II.J ARITHMETIC RATIO. 163 

last term is wanting. When that term is found, Their nature. 

the geometrical proportion becomes complete. 

In all such proportions, the first term is used as 

the divisor. Further, for every question in the 

Rule of Three, we have this clear and simple 

solution : viz. that, the unknown term or an- How solved. 

swer, is equal to the third term multiplied by 

the ratio of the first two. This simple rule, for 

finding the fourth term, cannot be given, unless ThisruJede - 

pends on the 

we define ratio to be the quotient of the second definition of 

Ratio. 

term divided by the first. Convenience, there- 
fore, as well as general analogy, indicates this as 
the proper definition of the term ratio. 

§ 170. Again, all authors, so far as I have Thisdefim- 
consulted them, are uniform in their definition 
of the ratio of a geometrical progression : viz. 
that it is the quotient which arises from divid- 
ing the second term by the first, or any other 
term by the preceding one. For example, in 
the progression 

2 : 4 : 8 : 16 : 32 : 64, &c, Example: 

all concur that the ratio is 2 ; that is, that it is in which 
the quotient wHch arises from dividing the sec- 
ond term by the first : or anv other term bv the 
preceding term. But a geometrical progression 
differs from a geometrical proportion only in 



used by all 
authors, in 
one case : 



agree. 



164 



MATHEMATICAL SCIENCE. 



[BOOK II. 



The same this: in the former, the ratio of any two terms 

should take . . 

piaceinevcry is the same; while in the latter, the ratio of the 

LTheyare m ' st anc * secon d is different from that of the sec- 
aii the same. ond and third There ^ fjgfaffc^ no essential 

difference in the two proportions. 

Why, then, should we say that in the propor- 
tion 

2 : 4 : : G : 12, 



the ratio is the quotient of the first term divided 

Example*. 

by the second ; while in the progression 

2 : 4 : 8 : 1G : 32 : 64, &c, 

the ratio is defined to be the quotient of the sec- 
ond term divided by the first, or of any term di- 
vided by the preceding term? 
wheroin As far as I have examined, all the authors 
who have defined the ratio of two numbers to 
be the quotient of the first divided by the sec- 
ond, have departed from that definition in the 
case of a geometrical progression. They have 
there used the word ratio, to express the quo- 
tient of the second term divided by the first, 
and this without any explanation of a change 
in the definition. 

Most of them have also departed from theii 
definition, in informing us that " numbers in- 
crease from right to left in a tenfold ratio/' in 



authors 
have depart- 
ed from their 
definitions : 



How used 
ratio. 



Other in- 
stances in 
which the 
definition of 



CHAP. II.] ARITHMETIC PROPORTION. 1G5 



which the term ratio is used to denote the quo- Ratio is not 

tient of the second number divided by the first. 

The definition of ratio is thus departed from, 

and the idea of it becomes confused. Such consequen- 
ces, 
discrepancies cannot but introduce confusion 

into the minds of learners. The same term 

should always be used in the same sense, and 

have but a single signification. Science does what science 

demands. 

not permit the slightest departure from this rule. 
I have, therefore, adopted but a single significa- 
tion of ratio, and have chosen that one to which The defini- 

tion adopted. 

all authors, so far as I know r , have given their 
sanction ; although some, it is true, have also 
used it in a different sense. 



§ 171. One important remark on the subject important 

r • i i t • i • Remark. 

oi proportion is yet to be made. It is this : 

Any two numbers which are compared togeth- Numbers 

T/r ' compared 

er, either by their difference or quotient, must must be of 

be of the same kind: that is, they must either kind 
have the same unit, as a basis, or be susceptible 
of reduction to the same unit. 

For example, we can compare 2 pounds with Examples 

G pounds : their difference is 4 pounds, and their Arithmetical 

ratio k the abstract number 3. We can also H^p^r" 

compare 2 feet with 8 yards : for, although the tion ' 
unit 1 foot is different from the unit 1 yard, still 
8 yards are equal to 24 feet. He" ce ; the differ- 



166 



MATHEMATICAL SCIENCE. [BOOK II 



ence of the numbers is 22 feet, and their ratio 
the abstract number 12. 
Numbers On the other hand, we cannot compare 2 dol- 

with different 

units cannot lars with 2 yards of cloth, for they are quantities 

be compared. 

of different kinds, not being susceptible of reduc- 
tion to a common unit. 
Abstract Simple or abstract numbers may always be 

numbers may 

be compared, compared, since they have a common unit 1 



SECTION IV. 



APPLICATIONS OF THE SCIENCE OF ARITHMETIC. 



§ 172. Arithmetic is both a science and an 
Arithmetic: art. It is a science in all that relates to the 
science, properties, laws, and proportions of numl>< 

The science is a collection of those connected 
science de- processes which develop and make known the 

fined. 

laws that regulate and govern all the operations 






performed on numbers. 



what the § 173. Arithmetic is an art, in this : the sci- 
ence lays open the properties and laws of num- 



forms. 



bers, and furnishes certain principles from which 



CHAP. II.] ARITHMETIC APPLICATIONS 167 

practical and useful rules are formed, applicable 
in the mechanic arts and in business transac- 
tions. The art of Arithmetic consists in the in what the 

i • • i ^ * r i t • r ..art consists. 

judicious and skiJul application of the princi- 
ples of the science; and the rules contain the 
directions for such application. 

§ 174. In explaining the science of Arithmetic, in explaining 
great care should be taken that the analysis of „ 1l , lMII „,j 
every question, and the reasoning by which the ■* 
principles are proved, be made according to the 
strictest rules of mathematical logic. 

Every principle should be laid down and ex- iioweach 
plained, not only with reference to its subsequent 



should be 
stated. 



use and application in arithmetic, but also, with 
reference to its connection with the entire mathe- 
matical science — of which, arithmetic is the ele- 
mentary branch. 

§ 175. That analysis of questions, therefore, what 
where cost is compared with quantity, or quan- que y°Z "" 
tity with cost, and which leads the mind of the 
learner to suppose that a ratio exists between 
quantities that have not a common unit, is, with- 
out explanation, certainly faulty as a process of 
science. 

For example : if two yards of cloth cost 4 dol- 

Example, 

lars, what will 6 yards cost at the same rate ? 



168 MATHEMATICAL SCIENCE. [BOOK II 

Analysis: Analysis. — Two yards of cloth will cost twice 
as much as 1 yard : therefore, if two yards of 
cloth cost 4 dollars, 1 yard will cost 2 dollars. 
Again : if 1 yard of cloth cost 2 dollars, 6 yards, 
being six times as much, will cost six times two 
dollars, or 12 dollars. 

satisfactory Now, this analysis is perfectly satisfactory to 
a child. He perceives a certain relation between 
2 yards and 4 dollars, and between 6 yards and 
12 dollars : indeed, in his mind, he compares 
these numbers together, and is perfectly satisfied 
with the result of the comparison. 

Advancing in his mathematical course., how- 
ever, he soon comes to the subject of propor- 
tions, treated as a science. He there iinds. 

Reason why greatly to his surprise, that he cannot compare 

it is defective. . ^ 

together numbers which have dinerent units ; 
and that his antecedent and consequent must be 
of the same kind. He thus learns that the whole 
system of analysis, based on the above method of 
comparison, is not in accordance with the prin- 
ciples of science. 
True What, then, is the true analysis ? It is this : 

analysis i , + * n , , . , . 

6 yards of cloth being 3 times as great as g 
yards, will cost three times as much : but 2 yards 
cost 4 dollars; hence, 6 yards will cost 3 tii 
4, or 12 dollars. If this last analvsis be not 

More scien- " 

mc - as simple as the first, it is certainly mote strictly 



CHAP. II. J ARITHMETIC APPLICATIONS. 169 



scientific ; and when once learned, can be ap- its 
plied through the whole range of mathematical 
science. 



§ 176. There is yet another view of this ques- Reasons m 

... , . c favor of the 

tion which removes, to a great degree, it not fir?t analysi3> 
entirely, the objections to the first analysis. It is 
this : 

The proportion between 1 yard of cloth and 
its cost, two dollars, cannot, it is true, as the 
units are now expressed, be measured by a ratio, 
according to the mathematical definition of a 
ratio. Still, however, between 1 and 2, regard- 
ed as abstract numbers, there is the same relation x umber9 
existing as between the numbers 6 and 12, also mu f t ** re ~ 

° garded as ab- 

regarded as abstract. Now, by leaving out of stracl: 

view, for a moment, the units of the numbers, 

and finding 12 as an abstract number, and then The analysis 

. . . , ' then correct. 

assigning to it its proper unit, we have a correct 
analysis, as well as a correct result. 



§ 177. It should be borne in mind, that practi- How the 

roles of arith- 

cal arithmetic, or arithmetic as an art, selects meticare 
from all the principles of the science, the mate- 
rials for the construction of its rules and the 

proofs of its methods. As a mere branch of What 

practical knowledge, it cares nothing about the k p ™^f e 

forms or methods of investigation— it demands demands. 



170 MATHEMATICAL SCIENCE. [bOOKII. 

the fruits of them all, in the most concentrated 
Best mio cf and practical form. Hence, the best rule of art 4 
which is the one most easily applied, and which . 
reaches the result by the shortest process, is not 
always constructed after those methods which 
science employs in the development of its prin- 
ciples. 
Definition of For example, the definition of multiplication is, 
mU tion. 1Ca * ^ iat ^ ^ s tne P r °cess of taking one number, called 
the multiplicand, as many times as there arc 
what n do- un ^ s * n another called the multiplier. This defi- 
muiuis. nition, as one of science, requires two things. 
First 1st. That the multiplier be an abstract num- 

ber; and, 
Socond. ~dly. That the product be of the same kind as 

the multiplicand. 

These two principles are certainly correct, 
May bo an d relating to arithmetic as a science, are uni- 

differently yersa ]Jy true J} u j. are t | u » v U ni veisall V trilC, ill 
considered as J J J 

furnishing a the sense in which thev would be understood by 

rule of art 

learners, when applied to arithmetic as a mixed 
subject, that is, a science and an art ? Such an 
application would certainly exclude a large class 
of practical rules, which are used in the appli- 
cations of arithmetic, without reference to par- 
ticular units. 
Examples of p or example, if we have feet in length to be 

such 

applications, multiplied by feet in height, we must exclude the 



CHAP. II.] ARITHMETIC APPLICATIONS. 171 



question as one to which arithmetic is not appli- 
cable ; or else we must multiply, as indeed we 
do, without reference to the unit, and then assign 
a proper unit to the product. 

If we have a product arising from the three Wnen ^ e 

three factors 

factors of length, breadth, and thickness, the are lines. 
unit of the first product and the unit of the final 
product, will not only be different from each 
other, but both of them will be different from 
the unit of the given numbers. The unit of the The different 
given numbers will be a unit of length, the unit 
of the first product will be a square, and that of 
the final product, a cube. 

§ 178. Again, if we wish to find, by the best 0ther 

examples. 

practical rule, the cost of 467 feet of boards at 
30 cents per foot, we should multiply 467 by 
30, and declare the cost to be 14010 cents, or 
$140,10. 

Now, as a question of science, if you ask, can considered 
we multiply feet by cents ? we answer, certainly 
not. If you again ask, is the result obtained 
right ? we answer, yes. If you ask for the anal y- 
sys, we give you the following : 

1 foot of boards : 467 feet : : 30 cents : Answer. 

Now, the ratio of 1 foot to 467 feet, is the ab Ratio, 
stract number 467: and 30 cents being mulli- 



as a question 
of science. 



172 MATHEMATICAL SCIENCE. [BOOK lb 



plied by this number, gives for the product 14010 
cents. But as the product of two numbers is 

Product of 

two numerically the same, whichever number be used 

numbers, , . ,. , , 

as the multiplier, we know that 4G7 multiplied by 

30, gives the same number of units as 30 multi- 

The first rule plied by 467 : hence, the first rule for finding the 

correct. 

amount is correct. 

scientific in- § 17 ( .). I have given these illust rations to point 
ratigatkm: ^ ^ difference between a process of scientific 

Fractica investigation and a practical rule. 

rule : . 

The first should always present the ideal 
Their differ- the subject in their natural order and connect 
While the other should point OUl the best wax 
8isl3, obtaining a desired result. In the latter, the 
steps of the procesp may not conform to the or- 
der necessary for the investigation of principl 
but the correctness of tin- result must be suscepti- 
ble of rigorous proof. Much needless and un- 
profitable discussion has arisen on many of the 

111969 Of r 

orror. processes of arithmetic, from confounding a princi- 
ple of science with a rule of mere application. 



once: in 
what it con- 



CHAP. II.] 



ARITHMETIC- 



-ORDER. 



173 



SECTION V. 



METHODS OF TEACHING ARITHMETIC CONSIDERED. 



ORDER OF THE SUBJECTS. 



§ 180. It has been well remarked by Cousin, 
the great French philosopher, that " As is the 
method of a philosopher, so will be his system ; 
and the adoption of a method decides the destiny 
of a philosophy. 5 ' 

What is said here of philosophy in general, is 
eminently true of the philosophy of mathematical 
science ; and there is no branch of it to which 
the remark applies, with greater force, than to 
that of arithmetic. It is here, that the first no- 
tions of mathematical science are acquired. It 
is here, that the mind wakes up, as it were, to 
the consciousness of its reasoning powers Here, 
it acquires the first knowledge of the abstract — 
separates, for the first time, the pure ideal from 
the actual, and begins to reflect and reason on 
pure mental conceptions. It is, therefore, of the 
highest importance that these first thoughts be 
impressed on the rr.ind in their natural and proper 



Cousin. 

Method 

decides 

Philosophy. 



True in 
science. 



Why 
important in 
Arithmetic 



First 
thoughts 
should bo 

rightly 
impressed. 



174 MATHEMATICAL SCIENCE. 1_LJ00K II. 



Faculties to order, so as to strengthen and cultivate, at the 

be cultivated. . . . 

same time, the iaculties ot apprehension, discrim- 
ination, and comparison, and also improve the 
yet higher faculty of logical deduction. 

First point: § 181. The first point, then, in framing a 

course of arithmetical instruction, is to detcr- 

methodof mine the method of presenting the subject. Is 

presenting . 1 • • i 1 i • 

the subject, there any thing m the nature ot the subject it- 
self, or the connection of its parts, that points 
out the order in which these parts should be 
Laws of studied? Do the laws of science demand a 

science : . , 

what do particular order; or are the parts so loosely 
they require? connec ^ i as to rcn der it a matter of indiffer- 
ence where we begin and where we end ? A 
review of the analysis of the subject will aid us 
in this inquiry. 

Basis of the § 182. We have seen* that the science of 

science of 

numbers, numbers is based on the unit 1. Indeed, the 
in what the whole science consists in developing, explain- 

science 

consists. M*gi and illustrating the laws by which, and 

through which, we operate on this unit. There 

Three classes are three classes of operations performed on the 

of operations. 

unit one. 
1st. To i st rpo increase it according to certain scales, 

increase the ° 

unit. 



* Section 104. 



CHAP. II.] AR'.THMETIC INTEGER UNITS. 175 

forming the classes of simple and denominate 
numbers ; 

2d. To divide it in any way we please, form- 2d. To 
ing the decimal and vulgar fractions ; and, 

3d. To compare it with all the numbers which 3d. To com- 
come from it ; and then those numbers with each 
other. This embraces proportions, of which the 
Rule of Three is the principal branch. 

There is yet a fourth branch of arithmetic; Fourth 
viz. the application of the principles and of the 
rules drawn from them, in the mechanic arts Practical 

, . . .. . r , applications; 

and in the ordinary transactions of business. 

This is called the Art, or practical part, of these the 

art. 

Arithmetic. (See Arithmetical Diagram facing 
page 117.) 

Now, if this analysis be correct, it establishes Analysis 

, . . l-ii i • r • i • establishes 

the order in which the subjects ot arithmetic the order 
should be taught. 



INTEGER UNITS. 

§ 183. We begin first with the unit 1, and in- TT .. 

5 o ' Unit one 

crease it according to the scale of tens, forming fcc 1 * 3 *^ 

° ° according to 

the common system of integer numbers. We the scale of 

tens. 

then perform on these numbers the operations 

of the five ground rules ; viz. numerate them, Operations 

add them, subtract them, multiply and divide p 

them. 



176 MATHEMATICAL SCIENCE. [BOOK IJ 



Next increase We next increase the unit 1 according to the 
to varying varying scales of the denominate numbers, and 



scales. 



thus produce the system, called Denominate or 
Concrete Numbers ; after which we perform 
upon this class all the operations of the five 
ground rules. 



What order § 181. It may be well to observe here, that 
exact science t' ie ' aw ot exact science requires us to treat the 
require. denominate numbers first, and the numbers of 
the common system afterwards; for, the corn- 
Reason for mon system is but a variety of the class of de- 
nominate numbers ; viz. that variety, in which 
the scale is the scale of tens, and unvarying. 

Reason for But as $01716 knowledge of d suhjrcf fflUSi 

departing _. . 

fromit. a M generalization^ we are obliged to begin the 
subject of arithmetic with the simplest element 



F B A CT1 N A L UNITS. 

Divisions of § 185. We now pass to the second class of 
operations on the unit 1 ; viz. the divisions of 

General me- it. Here we pursue the most general method, 
and divide it arbitrarily ; that is, into any num- 
ber of equal parts. We then observe that the 

Method ao- division of it, according to the scale of tens, is 

bc^rone^s. but a particular case of the general law of di- 
vision. We then perform, on the fractional 



CHAP. II.] ARITHMETIC RATIO. 177 



units which thus arise, all the operations of the operations 

performed. 

five ground rules. 



RATI O, R RULE OF THREE. 

§ 186. Having considered the two subjects of subjects 

considered. 

integer and fractional units, we come next to 
the comparison of numbers with each other. 

This branch of arithmetic develops all the what this 

, r , , . r branch de- 

relative properties ot numbers, resulting from V eio P s. 
their inequality. 

The method of arrangement, indicated above, what the ar 

n x , . r . , . ranxrement 

presents all the operations oi arithmetic in con- d ^. 
nection with the unit 1, which certainly forms 
the basis of the arithmetical science. 

Besides, this arrangement draws a broad line What it fa* 
between the science of arithmetic and its ap- 
plications ; a distinction which it is very im- 
portant to make. The separation of the prin- Theory and 
ciples of a science from their applications, so sh ouidbe 
that the learner shall clearly perceive what is 8e P™ atelL 
theory and what practice, is of the highest im- 
portance. Teaching things separately, teaching Golden rules 
them well, and pointing out their connections, 
are the golden rules of all successful instruc- 
tion. 



§ 187. I had supposed, that the place of the 

12 



178 MATHEMATICAL SCIENCE. [ROOK II. 

Rule of Three, among the branches of arith- 
metic, had been fixed long since. But several 
Differences in authors of late, have placed most of the practi- 

arrangement; a . . 

cal subjects before this rule — giving precede] 
for example, to the subjects of Percentage, In- 
in what they terest, Discount. Insurance. &c. It is not easy 

consist. 

to discover the motive of this change. It is 

Ratio pwt of certain that the proportion and ratio of num- 
the Bctonce. 

bers arc parts of the science of arithmetic; and 

shouw pre- the properties of numbers which they unfold, 

cede applica- 
tions, arc indispensably ncc essary to a clear apprehen- 
sion of the principles from which the practical 
rules are constructed. 

We may, it is true, explain each example in 

Percentage, Interest, J discount, Insurance, &c, 

Canm.t well by a separate analysis. But this i< a matter 
change the _ . 

onlrr . ol much labor; and besides, does not conduct 
the mind to any general principle, on which 
all the operations depend. Whereas, if the Rule 
of Three be explained, before entering on the 

Advantages practical subjects, it. is a great aid and a pow- 

Jbiuing'the er ^ auxiliary in explaining and establishing 

* llloof all the practical rules. If the Rule of Three 

Three. r 

is to be learned at all, should it not rather 

precede than follow its applications? It is a 

great point, in instruction, to lay down a gen- 

The great era ] principle, as early as possible, and then con- 

principle of L l . 

instruction, nect with it, and with each other, all the subor 



CHAP. II.] ARITHMETIC PRACTICAL PART. 179 

dinate principles, with their applications, which 
flow from it. 

PRACTICAL PART. 

§ 188. We come next to the 4th division; A PP licati ^ 

of arithmetic 

viz. the applications of arithmetic. 

Under the classification w r hich we have indi- Whathaa 

been done. 

cated, all the principles of the science will have 
been mastered, when the pupil reaches this stage 
of his progress. His business will now be w T ith What 

remains to be 

the application of principles, and no longer in done, 
the study and development of the principles 
themselves. The unity and simplicity of this Unityofth0 

J k J classification. 

method of classification, may be made more ap- 
parent, by the aid of the arithmetical diagram 
which faces page 117. 

May we not then conclude that the subjects Howthesu b- 

J J jects should 

of arithmetic should be presented in the follow 7 - be presented, 
ing order : 

1st. All the methods of treating integer num- lst - integer 

numbers. 

bers, whether formed from the unit 1 according 

to the scale of tens, or according to varying 

scales ; 

2d. All the methods of treating fractional uni- 2d. Frac- 
tions, 
ties, whether derived from the unit 1 according 

to the scale of tens, or according to varying 

scales; 



180 MATHEMATICAL 6CIENCE. [BOOK II. 



3d. Rule of 3d. The proportion and ratios of numbers; 

Three. 

and, 
4th. Appiica- 4th. The applications of the science )f nuta- 



tions. 



bers to practical and useful objects. 



OBJECTIONS TO THIS CLASSIFICATION ANSWER 2D. 



Twoobjec- § 189. It has been urged that Common or Vul- 
.tionstothis Fractions should be placed "immediately 

classification. ~ l 

after Division, for two reasons" 

"First, they arise from division, being in fact 

First 

unexecuted division/' 
"Second, in Reduction and the Compound 

Second. 

Kulcs.it is often necessary to multiply and divide 

fractions, to add and subtract them, also to carry 
for them, unless perchance the examples are con- 
structed for the occasion, and with special refers 
ence to avoiding these difficulties." 

These are all. These, I believe, are all the objections that 
have been, or can be urged against the classifi- 
cation which I have suizize>ted. I give them in 
Given in full, full, because I wish the subject of arrangement 
to be fully considered and discussed. It should 
what be our main object to get at the best possible 
should be system f classification, and not to waste our 

our object. J 

efforts in ingenious arguments in the support of 
To be con- a favorite one. We will consider these objec 

•idered se- . . 

parateiy. tions separately 



CHAP- II. J ARITHMETIC OBJECTIONS. 181 



It is certainly true, that fractions " arise from Fractions 

,...,,,.. . arise from di- 

division, but it is as certainly not true, that they vision. 
are " unexecuted divisions ;" and this last idea 
has involved the subject in much perplexity and 
difficulty. 

The most elementary idea of a fraction, arises The element- 

ary idea is 

from the division of a single thing into two equal obtained by 

parts, each of which is called a half. Now, we u^divfekHi: 

get no idea of this half unless we consider the 

division perfected. And indeed, the method of 

teaching shows this. For, we cannot impress 

the idea of a half on the mind of a child, until Example; 

we have actually divided in his presence the 

apple (or something else regarded as a unit), 

and exhibited the parts separately to his senses ; 

and all other fractions must be learned by a like 

reference to the unit 1. Hence, we can form no And not 

r r . . . otherwise. 

notion of a fraction, except on the supposition of 
a perfected division. 

If the term, " unexecuted division/' applies to "Unexecuted 

. division ?, does 

the numerator of the expression, and not to the not apply to 
unit of the fraction, the idea is still more in- ien t o™ er 
volved. For, nothing is plainer than that we 
can form no distinct notion of a result, so long 
as the process on which it depends cannot be 
executed. The vague impression that there is 

Thatafrac- 

something hanging about a fraction that cannot turn cannot 
be quite reached, has involved the subject in a r€ achcd,has 



I 82 M A T II E II ATICAL SCIENCE. [BOOK II. 

or^M.mod mysterious terror; and the boy approaches it 
difficulty. . . 

with the same ieelmg which a manner dor- 
rocky and dangerous coast, of which lie has 
neither map nor chart to guide him. But | 
sent to the mind of the pupil the distinct idea, 
that a traction is one or more equal parts of 
unity, and that every such part is & perfect whole, 
hating a certain relation to the thing from which 

it was derived) and all the mist b cleared away, 
and his mind divides the unit into any number 
of equal parts, with the same facility as the knife 
divides the apple. 
Form the The form of expression for a fraction, and tor 

same as for ..... • • i 1 i i 

an unexecuted division, is indeed the same, nut 



Every frac- 
tion has ft 

flxed relation 

to unity. 



bd unexecu- 
ted division 



the interpretation of this expression, as used for 

one or the other, is entirely different In our 

Asi-nmay common Language, the same word is not al- 

express dn> 

ferent thlnp. wavs the sign of the same idea; and in science, 

the same symbol often expresses very different 

things, 

riMniii For example, -?. as an expression in fractions, 

illustrating . . . , . , , . 

heseprinci- means, that something regarded as a whole lias 

plos * been divided in 7 equal parts, and that 3 of those 

parts are taken. As a result ofdivision.it means 

that the integer number 3 is to be divided into 

what cannot 7 equal parts. Now, it cannot be assumed, as a 

be assumed. 

self-evident fact, that three of the parts of the 
first division are equal to 1 part of the second; 



CHAP. II.] ARIT HM ETIC OBJECTIONS. 



183 



and if this fact be made the basis of a system 

of fractions, the mind of a child will go through Thebaslsof 

° every system 

that system in the dark. The basis of every sys- should be an 

elementary 

tern should be an elementary idea, idea. 



§ 190. The second objection, as far as it goes, 
is valid. In all the tables of denominate num- 
bers, fractions occur five times ; viz. twice in 
Long Measure, where 5|- yards make 1 rod, and 
69j statute miles 1 degree ; once in Cloth Mea- 
sure, where 2\ inches make 1 nail; once in 
Square Measure, where 30j square yards make 
1 square rod ; and once in Wine Measure, where 
31^ gallons make 1 barrel. Now, it were a little 
better, if these tables had been constructed with 
integer units. But it should be borne in mind, 
that the first notions of fractions are given either 
by oral instruction, or learned from elementary 
arithmetics. Most of the leading arithmetics 
are, I believe, preceded by smaller works. These 
are designed to impart elementary ideas of num- 
bers, so as not to embarrass the classification of 
subjects when the scholar is able to enter on a 
system. Now, the most elementary of these 
works conducts the pupil, in fractions, far be- 
yond the point necessary to understand and 
manage ail the fractions which appear in the 
tables of denominate numbers; and hence, there 



Second objec- 
tion valid ; 



But of no 
great weight. 



Reasons. 



Design of 
smaller 
works. 



Fractions are 

partially 

taught in the 

elementary 

works ; 



184 MATHEMATICAL SCIENCE. [BOOK II. 



May then be is no reason, on that account, to depart from a 

used. 

classification otherwise desirable. 



OBJECTIONS TO THE NEW METHOD. 

§ 191. Having examined the objections that 
have been urged against that system of classifi- 
cation of the subjects of arithmetic, which has 



Objections to 



the new me- appeared to me most in accordance with the 

iiiod consid- ... r 

ered. ] >nnci pies 01 science, I >hall now point out N 

of the difficulties to be met with in the adoption 

of the method proposed as a substitute. 
First obj.c- 1st. Thai method separates the simple and de- 
nominate numbers, which, in their general form- 
ation, differ from each other only in the » 

by which we pass from one unit of value to an- 
other. 

Secondobjec- 2d. By thus separating these numbers, it be- 

tion. i • -r i • l 

comes more ditlieult to point out their connec- 
tion and teach the important fact, that in all 
their general properties, and in all the opera- 
tions to be performed upon them, they differ 

from each other in no important particular. 

Thud objec- 3d. By placing the denominate numbers after 
Vulgar Fractions, all the principles and rules in 

limitation of Fractions are limited in their application to a 
the mies. s { n g] e rf ass f fractions ; viz. to those fractions 
which have the same unit. 



CHAP. II.] ARITHMETIC OBJECTIONS. 185 



. For example, the common rule for addition Examples; 

• •• i showing thii 

of fractions, under this classification, is, in sub- 
stance, the following : " Reduce the fractions to 
a common denominator; add their numerators, Rule; not 

general. 

and place the sum over the common denomi- 
nator!' 

As the subject of denominate numbers has Have not ye» 

" considered 

not yet been reached, no allusion can be made fractions 
to fractions having different units. If the learn- a ™ g . er * 

o JJ ent units ; 

er should happen to understand the rule literally, 
he would conclude that, the sum of all fractions 
having a common denominator is found by sim- 
ply adding their numerators and placing the The rules 

therefore ap- 

sum over the common denominator. But this piytoone 
cannot, of course, be so, since £ of a £ and f ^oniy. 
of a shilling make neither one pound nor one 
shilling. 

What appears to me most objectionable in Greatest ob- 
this method, is this : it fails to present the im- 
portant fact, that no two fractions can be blend- 
ed into one, either by addition or subtraction, 
unless they are parts of the same unit, as weJ 
as like parts. 

By this method of classification most of the This method 
difficult questions which arise in fractions are tionavoid3 
avoided, or else the subject must be resumed thedifficult 

J questions. 

after denominate numbers, and that class of 



questions treated separately 



186 MATHEMATICAL SCIENCE. [BOOK II. 



whatthoy The class of questions to which I refer, em- 
are. 

braces examples like the following: 

Add j of a day, j\ of an hour and f of a sec- 
ond together. 

It is certainly true that a boy will make mar- 
vellous progress in the text-hook, if you limit 
The subject him to those examples in which the fractions 

c.-isily <lis- , 

posed of, but have a common unit. But, will he ever un- 
derstand the science of fractions unless his mind 
be steadily and always turned to the unit of the 

fraction, as the bask? Will he understand the 
value of one equal part. s<> as to compare and 

unite it with another equal part, unless be first 
apprehends, clearly, the units from which those 
parts were derived f 

Lutotyeo- lth. By placing the Denominate Numbers be- 

tion staled. T . , „ . 

tween Vulgar and Decimal tractions, the L r en- 

eral subject of fractional arithmetic is broken 

into fragments. This arrangement makes it dif- 

Difficuityof ficult to realize that these two systems of num- 

connection of bers differ from each other in no essential par- 

ns. t j cu ] ar . t ] iat t | 10V are ^ )(> t h formed from the unit 

one by the same process, with only a slight m 

ification of the scale of division. 



CHAP. II.] ARITHMETIC LANGUAGE. 187 



ARITHMETICAL LANGUAGE. 

§ 192. We have seen that the arithmetical al- Arithmetical 

alphabet. 

phabet contains ten characters.* From these 
elements the entire language is formed; and we 
now propose to show in how simple a manner. 

The names of the ten characters are the first Names of the 

characters. 

ten words of the language. If the unit 1 be 

added to each of the numbers from 1 to 10 in- First ten 

elusive, we find the first ten combinations in tions. 

arithmetic. f If 2 be added, in like manner, 

we have the second ten combinations ; adding second ten, 

and so on for 

3, gives us the third ten combinations ; and so others, 
on, until we have reached one hundred com- 
binations (page 123). 

Now, as we progressed, each set of combina- Each set giv- 
ing one addi- 
tions introduced one additional word, and the uonaiword. 

results of all the combinations are expressed by 

the words from two to twenty inclusive. 



§ 193. These one hundred elementary com- au that need 
binations, are all that need be committed to ted tome- 



memory; for, every other is deduced from them. 
They are, in fact, but different spellings of the 
first nineteen words which follow one. If we ex- 
tend the words to one hundred, and recollect that 

* Section 114. t Section 116. 



mory. 



188 MATHEMATICAL SCIENCE. [BOOK II. 



at one hundred, we begin to repeat the numbers, 
words to be we see that we have but one hundred words to 

remembered , 

for addition, be remembered for addition; and ot these, all 
only ten above ten are derivative. To this number, 

words primi- 
tive, must of course be added the few words which 

express the sums of the hundreds, thousands, &c. 

Subtraction: § 191. In Subtraction, we also find one hun- 
dred elementary combinations; the results of 
which arc to be read.* The llts, and all 

KumberoT the liuinlnTs en ij >1< wed in obtaining them, are 

words. 

expressed by twenty words. 

Multiplier- § 105. In Multiplication (the table being car- 
ried to twelve), we have one hundred and forty- 
four elementary combinations,! and fifty-nine 

Numbrrof separate words (already known) to express the 
results of these combinations. 

Division: § 196. In Division, also, we have one hundred 
„ , . and forty-four elementary combinations.! but 

Number of J J 

words. use on ]y twelve words to express their results. 

Four hun- 

. < ' ; " 11 '""\ § 197. Thus, we have four hundred and eigfe 
elementary ty-eight elementary combinations. The results 

combina- 
tions, of these combinations are expressed by one hun- 

vvordsused: ^ ve( ^ WO rds ; viz. nineteen in addition, ten in sub- 

19 in addi- 

tion ' traction, fifty-nine in multiplication, and twelve 

10 in subtrac 

lion, ' 

aiinmuiu- # Action 120. \ Section 122. \ Section I2S 

plication, T 



CHAP. II.] ARITHMETIC LANGUAGE. 189 



in division. Of the nineteen wcrds which are i2in division 
employed to express the results of the combina- 
tions in addition, eight are again used to express 
similar results in subtraction. Of the fifty-nine 
which express the results of the combinations 
in multiplication, sixteen had been used to ex- 
press similar results in addition, and one in 
subtraction ; and the entire twelve, which ex- 
press the results of the combinations in division, 
had been used to express results of previous 
combinations. Hence, the results of all the ele- 
mentary combinations, in the four ground rules, 
are expressed by sixty-three different w T ords ; and sixty-three 
they are the only w r ords employed to translate wonbinan. 
these results from the arithmetical into our com- 
mon language. 

The language for fractional units is similar Language 

7 -p. r ^ tne same f° r 

in every particular. r>y means ot a language fractions, 
thus formed we deduce every principle in the 
science of numbers. 

§ 198. Expressing these ideas and their com- 
binations by figures, gives rise to the language Language of 

c ., . -^ ! •iri«i arithmetic: 

ot arithmetic. By the aid ot this language we 

not only unfold the principles of the science, its value and 

but are enabled to apply these principles to 

every question of a practical nature, involving 

the use of figures. 



190 MATHEMATICAL SCIENCE. [bOOKIK 

But few § 199. There is but one further idea to be 

combinations . . . . . . . r 

which presented : it is this, — that there are very lew 
change the com bi na ti ons ma de among the figures, which 

signification o ° 

of the figures, change, at all, their signification. 

Selecting any two of the figures, as 3 and 5, 
„ . for example, we see at once that there are but 

Examples. 1 J 

three ways of writing them, that will at all 
phange their signification. 

rirst: First, write them by the side of each j 3 .">. 

other ) 5 3. 

second: Second, write them, the one over > 

the other ) J. 

Tiiird. Third, place a decimal point before J .8, 

each j .."). 

Now, each manner of writing a differ- 

ent signification to both the figures. Use, how- 
Learathe ever, has established that signification, and w 



e 



uguajpty know it, as soon as we have learned the lan- 



guage. 



We have thus explained what we mean by 

the arithmetical language. Its grammar em- 
its grammar: braces the names of its elementary signs, or 
Alphabet- Alphabet, — the formation and number of its 
their uses, words, — and the laws by which figures are con- 
nected for the purpose of expressing ideas. We 
feel that there is simplicity and beauty in this 
system, and hope it may be useful. 



CHAP. II.] ARITHMETIC DEFINITIONS. 191 



NECESSITY OF EXACT DEFINITIONS AND TER3IS. 

§ 200. The principles of every science are Principles of 
a collection of mental processes, having estab- 
lished connections with each other. In every 
branch of mathematics, the Definitions and Definitions 

and terms : 

Terms give form to, and art* he signs of, cer- 
tain elementary ideas, which are the basis of 
the science. Between any term and the idea 
which it is employed to express, the connection 
should be so intimate, that the one will always 
suggest the other. 

These definitions and terms, when their sig- when once 
nifications are once fixed, must always be used always be 



used in the 
same sense. 



in the same sense. The necessity of this is most 
urgent. For, "in the whole range of arithmetical 
science there is no logical test of truth, but in Reason. 
a conformity of the reasoning to the definitions 
and terms, or to such principles as have been 
established from them." 

§ 201. With these principles, as guides, we Definition* 
propose to examine some of the definitions and examined, 
terms which have, heretofore, formed the basis 
of the arithmetical science. We shall not con- 
fine our quotations to a single author, and shall 
make only those which fairly exhibit the gen- 
eral use of the terms 



192 MATHEMATICAL SCIENCE. [BOOK II. 

It is said, 
Number de- "Number signifies a unit, or a collection of 

fined. . ., 

units. 
How "The common method of expressing numbers 

exnrcs'fd. 

is by the Arabic Notation. The Arabic method 
employs the following ten characters, or figur 
&c. 
Namesofthe " The first nine are called significant figures, 

characters. . 

because each one always has a value, or denotes 
some number.'' 

And a little further on we have, 
Figures hm "The different values which figures have, 

▼ alucs. 

called simple and local values. 

The definition of Number i< clear and cor- 
Number reel It is a general term, comprehending aj 

rightly de- . , ... . . . 

fined: ™° phrases which are used, to express, either 
separately or in connection, one or more things 

Also figures, of the same kind. So, likewise, the definition 
of figures, that thev are characters, \< also right. 

Deflnitionde- But mark how soon these definitions are de- 
parted from. The reason given why nine of the 

figures are called significant is. because '-each 
one always has a value, or denotes some num- 
ber." This brings us directly to the question. 
Has a figure whether a figure has a value : or, whether it is 
a mere representative of value. Is it a number 
or a character to represent number? Is it a 

It is merely . _ . _ , 

a character: quantity or symbol ! Jt is denned to be a char- 



CHAP. II.] ARITHMETIC DEFINITIONS. 193 

acter which stands for, or expresses a number. 
Has it any other signification ? How then can 
we say that it has a value — and how is it possi- Has uo value 
ble that it can have a simple and a local value ? 
The things which the figures stand for, may 
change their value, but not the figures them- 
selves. Indeed, it is very difficult for John to 
perceive how the figure 2, standing in the sec- but stands 

, , . _ for value. 

ond place, is ten times as great as the same fig- 
ure 2 standing in the first place on the right! 
although he will readily understand, w 7 hen the 
arithmetical language is explained to him, that 
the unit of one of these places is ten times as unit of place, 
great as that of the other. 

§ 202. Let us now 7 examine the leading defi- Leading dea 
nition or principle which forms the basis of the 
arithmetical language. It is in these words : 

"Numbers increase from right to left in a of number. 
tenfold ratio ; that is, each removal of a figure 
one place towards the left, increases its value 
ten times" 

Now, it must be remembered, that number Does not 
has been defined as signifying " a unit, or a thTdefini- 
collection of units." How, then, can it have a tionbefor * 

given. 

right hand, or a left ? and how can it increase 
from right to left in a tenfold ratio?" The 
explanation given is, that "each removal of a 

13 



194 



MATHEMATICAL SCIENCE. 



[book D 



Explanation. 



Increase of 
numbers has 
no connection 

with figures. 



Ratio. 



"Tenfold 
ratio :" 



Four leading 
notions of 
numbers. 



First 



Second. 



Third. 



Fourth. 



figure one place towards the left, increases its g 
value ten times!' 

Number, signifying a collection of units, must 
necessarily increase according to the law by 
which these units are combined; and that law 
of increase, whatever it may be, has not the 
slightest connection with the figures which are 
used to express the numbers. 

Besides, is the term ratio (yet undefined), 
one which expresses an elementary idea? And 
is the term, a " tenfold ratio" one of sufficient 
simplicity for the basis of a system ? 

Does, then, this definition, which in substance 
is used by most authors, involve and carry to 
the mind of the young learner, the four leading 
ideas which form the basis of the arithmetical 
notation ? viz. : 

1st. That numbers are expressions for one or 
more things of the same kind. 

2d. That numbers are expressed by certain 
characters called figures ; and of which there 

are ten. 

3d. That each figure always expresses as 
many units as its name imports, and no more. 

4th. That the kind of thing which a figure 
expresses depends on the place which the figure 
occupies, or on the value of the units, indicated 
in some other way. 






CHAP. II.] ARITHMETIC DEFINITION S . 



195 



Place is merely one of the forms of language mace; 
by which we designate the unit of a number, its office, 
expressed by a figure. The definition attributes 
this property of place both to number and fig- 
ures, while it belongs to neither. 



§ 203. Having considered the definitions and 
terms in the first division of Arithmetic, viz. in 
notation and numeration, we will now pass to Definitional! 

, , ait* Addition: 

the second, viz. Addition. 

The following are the definitions of Addition, 
taken from three standard works before me : 

" The putting together of two or more num- First. 
bers (as in the foregoing examples), so as to 
make one whole number, is called Addition, and 
the whole number is called the sum, or amount." 

" Addition is the collecting of numbers to- second, 
gether to find their sum." 

" The process of uniting two or more num- Third. 
bers together, so as to form one single number, 
is called Addition." 

" The answer, or the number thus found, is 
called the sum, or amount." 

Now, is there in either of these definitions Defects, 
any test, or means of determining when the 
pupil gets the thing he seeks for, viz. "the sum 
of two or more numbers ?" No previous defi- Reason, 
nition has been given, in either work, of the 



196 MATHEMATICAL SCIENCE. [BOOK II. 

term sum. How is the learner to know what 
he is seeking for, unless that thing be defined ? 
No prm- Suppose that John be required to find the sum 

ciple as a 

standard, of the numbers 3 and 5, and pronounces it to 
be 10. How will you correct him, by showing 
that he has not conformed to the definitions and 
rules? You certainly cannot, because you have 
established no test of a correct process. 

But, if you have previously defined sum to be 
a number which contains as many units as th 
are in all the numbers added : or, if you say, 

correct defl- "Addition is the process of uniting two OT 
more numbers, in such a way, that all the units 
which they contain may be expressed by a sin- 
gle number, called the sum, or sum total;" jroa 
will then have a test for the correctness of the 

Givesatest. process of Addition; viz. Does the number, 
which you call the sum, contain as many units 
as there are in all the numbers added ? The 
answer to this question will show that John is 



Definitions of § 204. I will now quote the definitions of 
Fractions from the same authors, and in the 
same order of reference. 
First. " We have seen, that numbers expressing whole 

things, are called integers, or whole numbers ; 
but that, in division, it is often necessary to 



CHAP. II.] ARITHMETIC DEFINITIONS. 197 

divide or break a whole thing into parts, and 
that these parts are called fractions, or broken 
numbers/' 

" Fractions are parts of an integer." second. 

" When a member or thing is divided into Third. 
equal parts, these parts are called Fractions. " 

Now, will either of these definitions convey 
to the mind of a learner, a distinct and exact 
idea of a fraction ? 

The term "fraction," as used in Arithmetic, Term fraction 

, r . . defined. 

means one or more equal parts ot something 
regarded as a whole : the parts to be expressed 
in terms of the tiling divided considered as a 
unit. There are three prominent ideas which ideas 

exDressed • 

the mind must embrace : 

1st. That the thing divided be regarded as a First, 
standard, or unity ; 

2d. That it be divided into equal parts ; second. 

3d. That the parts be expressed in terms of Third, 
the thing divided, regarded as a unit. 

These ideas are referred to in the latter part Thedefini- 
of the first definition. Indeed, the definition ^nedf 11 
would suggest them to any one acquainted with 
the subject, but not, we think, to a learner. 

In the second definition, neither of them is isafrac- 
hinted at. Take, for example, the integer num- anln^er* 
ber 12, and no one would say that any one part 
of this number, as 2, 4, or 6. is a fraction, 



1 i)8 M A T II E M A T I C A L SCIENCE. [BOOK II. 



Third The third definition would be perfectly accu- 

definition ; . . r . ., 

rate, by inserting after the word M thing, the 
words, "regarded as a whole/' It very clearly 
expresses the idea of equal parts, but doei not 
in^hatde- present the idea strongly enough, that the thing 
divided must be regarded as unit}', and that the 
parts must be expressed in terms o( this unity. 

§ -jo."). I have thus given a few examples, illus- 
v Mityoi trating the necessity of accurate definitions and 

terms. Nothing further need be added, except 

the remark, that terms should always be used in 
the same sense, precisely, in which they are de- 
fined, 
otyecuoa To some, perhaps, these distinctions may ap* 

of thought l ,(>ar OTcr-niceip and matters of little moment 
■ndiugaage. ^ m;iv ^ SU j,j )()Sr( | (^ ., general impression, 

imparted by a language reasonably aeein 
will suffice very well ; and that it is hardly 

worth while to pause and weigh words on a 

nicely-adjusted balance. 

Any such notions, permit me to say, will lead 
to fatal errors in education. 
Definitions in It is in mathematical science alone that words 
are the signs of exact and clearly-defined ideas. 
It is here only that we can see, as it were, the 
very thoughts through the transparent words by 
which they are expressed. If the words of the 



CHAP. II.] ARITHMETIC SUBJECTS. 199 



definitions are not such as convey to the mind Must be 

exact to 

of the learner, the fundamental ideas of the reason cor- 
science, he cannot reason upon these ideas ; 
for, he does not apprehend them ; and the great 
reasoning faculty, by which all the subsequent 
principles of mathematics are developed, is en- 
tirely unexercised.* 

It is not possible to cultivate the habit of cannot other. 

.... . . wise cultivate 

accurate thinking, without the aid and use of habits of 
exact language. Xo mental habit is more use- 
ful than that of tracing out the connection be- 
tween ideas and language. In Arithmetic, that 
connection can be made strikingly apparent, connection 
Clear, distinct ideas — diamond thoughts — may won j 3a nd 
be strung through the mind on the thread of til0u:?ht5 . ia 

° <~> antninetic. 

science, and each have its word or phrase by 
w r hich it can be transferred to the minds of 
others. 



HOW SHOULD THE SUBJECTS BE PRESENTED? 

§ 200. Having considered the natural connec- what 
tion of the subjects of arithmetic with each considered, 
other, as branches of a single science, based on 
a single unit; and having also explained the 
necessity of a perspicuous and accurate Ian- 



Section 200. 



200 iM A T H E M ATICAL SCIENCE. [BOOK II- 

How ought guage ; we come now to that important inquiry, 

the subjects 

tobepre- How ought those subjects to be presented to the 
bent ' mind of a learner ? Before answering this ques- 
Two objects tion, we should reflect, that two important ob- 
arithmetic°: jects should be soqght after in the study of arith- 
metic : 
nnt 1st. To train the mind to habits of clear; 

quick, and accurate thought — to teach it to 
apprehend distinctly — to discriminate closely — 
to judge truly — and to reason correctly; and, 
Bemd. 2d« To in abundance,, that practical 

knowledge of the Use of figures, in their va- 
rious applications, which shall illustrate the - 
Artofarith- king fact, that the art of arithmetic is the i 

important art of civilized lift — being, in fact t 
the foundation of nearly all the oth 

now r.rst im- § 207. It is certainly true, that most, if not 

mdJ" aB tno elementary notions, whether abstract <>r 

practical — that is, whether they relate tO the 

science or to the art of arithmetic, must be 

made on the mind by means of sensible obje< 

Because of this fact, many have supposed that 

is reason- the processes of reasoning are all to be con- 

iD ductedby D * ducted ky the same sensible objects; and that 

sensible ever y abstract principle of science is to be de- 
objects? J r r 

veloped and established by means of sofas, 
chairs, apples, and horses. There seems to be 



CHAP 



it.] 



ARITHMETIC- 



-SUBJECTS. 



201 



an impression that because blocks are useful 
aids in teaching the alphabet, that, therefore 
they can be used advantageously in reading 
Milton and Shakspeare. This error is akin to 
that of attempting to teach practically, Geog- 
raphy and Surveying in connection with Geom- 
etry, by calling the angles of a rectangle, north, 
south, east, and west, instead of simply designa- 
ting them by the letters A, B, C, and D. 

This false idea, that every principle of sci- 
ence must be learned practically, instead of 
being rendered practical by its applications, has 
been highly detrimental both to science and art. 

A mechanic, for example, knowing the height 
of his roof and the width of his building, wishes 
to cut his rafters to the proper length. If he 
calls to his aid the established, though abstract 
principles of science, he finds the length of his 
rafter, by the well-known relation between the 
hypothenuse and the two sides of a right-angled 
triangle. If, however, he will learn nothing ex- 
cept practically, he must raise his rafter to the 
roof, measure it, and if it be too long cut it off, 
if too short, splice it. This is the practical way 
of learning things. 

The truly practical way, is that in which skill 
is guided by science. 

Do the principles above stated find any appli- 



Scnsible 
objects useful 
in acquiring 
the simplest 

elements : 



Error 

of carrying 

them 

beyond. 



False idea : 



Its effects. 



Example 

of the appli- 
cation of 
an abstract 

principle: 



Of learning 
practically. 



True 
practwaL 



202 MATHEMATICAL SCIENCE. [BOOK If. 



Can 



cation in considering the question, How should 
arithmetic be taught? Certainly they do. if 
beapp.e. ar j t k met j c j 3e both a sc j e nce and an art, it 

should be so tauuht and so learned. 



*C5 



PriMptw §208. The principles of every science are get** 

****** era i all( i abstract truths. They an mere id 
what primarily acquired through the s by experi- 

ence, and generalized by processes af reflection 
wi- and reasoning; and when understood, are certain 

'" "** guides in ever}' 6a6e tO which they are applicable. 

If W€ choose tO do without them, we may. But 

is it wise to turn our beadti from the guide-boards 
and explore every road that 6pens before i 

Now. in the Study of arithmetic those princi- 
ples of science, applicable to classes of 

When should always be taught at the earliest possible 
they should moment. The mind should never be forced 

be taught t i irouir | 1 a ^ng srnr< () f examples, without es- 

n*e methods planation. One or two examples should alw 

precede the statement of an .abstract principle, 

or the laying down of a rule. SO as to make the 

anguage of the principle or rule intelligible. 

But to carry the learner forward through a 

iv.ncipw series of them, before the principle on which 

tobe^mpics- ^^ depend has been examined and stated, is 
forcina the mind to advance mechanically — it 
is lifting up the rafter to measure it, when its 






CHAP. II.] ARITHMETIC TEXT-BOOKS. 203 

exact length could be easily determined by a 
rule of science. 

As most of the instruction in arithmetic must Books: 
be given with the aid of books, we feel unable 
to do justice to this branch of the subject with- Necessity 
out submitting a few observations on the nature of them. 3 
of text-books and the objects which they are in- 
tended to answer. 



TEXT -BOOKS. 

§ 209. A text-book should be an aid to the Text-book 
teacher in imparting instruction, and to the 
learner in acquiring knowledge. 

It should present the subjects of knowledge what it 
in their proper order, with the branches of each 
subject classified, and the parts rightly arranged. 
No text-book, on a subject of general knowledge, selection 
can contain all that is known of the subject on necessary, 
which it treats ; and ordinarily, it can contain 
but a very small part. Hence, the subjects to 
be presented, and the extent to which they are Difficultly 
to be treated, are matters of nice discrimination 
and judgment, about which there must always 
be a diversity of opinion. 

§ 210. The subjects selected should be leading subjects: 
ones, and those best calculated to unfold, ex- 



201 



MATHEMATICAL SCIENCE. 



[book M. 



How 
presented, 



plain, and illustrate the principles of the science. 
They should be so presented as to lead the mind 
to analyze, discriminate, and classify ; to see 
each principle separately, each in its combina- 
tion with others, and all, as forming an harmo- 
nious whole. Too much care cannot be be- 
stowed in forming the suggestive method oj 
arrangement: that is, to place the ideal and 
principles ID such a connection, that each step 
Kcasontor. shall prepare t/ie mind of the learner for the next 
i/t order. 



Suggestive 
method : 



Object 
of a text- 
book: 



Nature; 



Useless 
detail ; 



Should 
not bo his- 
torical. 



§*2ll. A text-book should he constructed for 
the purpose <>f furnishing the learner with the 

keys of knowledge. It should point out, explain, 

and illustrate by examples, tii'' methods of in- 
vestigating and examining subjects, but should 

leave the mind of the learner Tree from the re* 
straints of minute detail. To III 1 a book with 
the analysis of simple questions, which any child 
can solve in his own way, is to constrain and 
force the mind at the very point where it is i 
pable of self-action. To do that for a pupil, 
which he can do for himself, is most unwise. 

§ 212. A text-book on a subject of science 
should not be historical. At first, the minds of 
children are averse to whatever is abstract, be- 









CHAP. II.] ARITHMETIC TEXT-BOOKS. 



205 



cause what is abstract demands thought, and Reasons, 
thinking is mental labor from which untrained 
minds turn away. If the thread of science be 
broken by the presentation of facts, having no 
connection with the argument, the mind will 
leave the more rugged path of the reasoning, 
and employ itself with what requires less effort 
and labor. 

The optician, in his delicate experiments, ex- illustration, 
eludes all light except the beam which he uses : 
so, the skilful teacher excludes all thoughts 
excepting those which he is most anxious to 
impress. 

As a general rule, subject of course to some 
exceptions, but one method for each process one imtk i 
should be given. The minds of learners should 
not be confused. If several methods are given, Reasons 
it becomes difficult to distinguish the reasonings 
applicable to each, and it requires much knowl- 
edge of a subject to compare different methods 
with each other. 



§ 213. It seems to be a settled opinion, both 
among authors and teachers, that the subject of 
arithmetic can be best presented by means of 
three separate works. For the sake of distinc- 
tion, we will designate them the First, Second, 
and Third Arithmetics. 



How the 
subject is 
dirided. 



It* 

importance. 



20G MATHEMATICAL SCIENCE. [BOOK II. 

We will now explain what we suppose to be 
the proper construction of each book, and the 
object for which each should be designed. 

FIRST ARITHMETIC. 

First § 211. This book should give to the mind 

Arithmetic : 

its first direction in mathematical science, and 
its first impulse in intellectual development. 
Hence, it is the most important book of the 
series. Here, the faculties of apprehension, dis- 
crimination, abstraction, classification and com- 
parison, are brought first into activity. N?Wi 
now to cultivate and develop these faculties rightly, 

the Hibjects 

m.^tbe we must, at first, present every new idea by 
p,( ' "" means of a sensible object, and then immedi- 
ately drop the object and pass to the abstract 
thought. 

Order We must also present the ideas consecutively; 

oft he Idem. 

that is, in their proper order; and by the mere 
method of presentation awaken the comparative 
and reasoning faculties. Hence, every lesson 
should contain a given number of ideas. The 
construction ideas of each lesson, beginning with the first, 

•ft he lessons. . . 

should advance in regular gradation, and the 
lessons themselves should be regular steps in 
the progress and development of the arithmeti- 
cal science. 






CHAP. II.] ARITHMETIC TEXT-BOOKS. 



207 



§ 215. The first lesson should merely contain 
representations of sensible objects, placed oppo- 
site names of numbers, to give the impression 
of the meanings of these names : thus, 

One * 

Two * * 

Three * * * 

&c. &c. 

And with young pupils, more striking objects 
should be substituted for the stars. 

In the second lesson, the words should be re- 
placed by the figures : thus, 

1 * 

2 * * 

&c. &c. 

In the third lesson, I would combine the ideas 
of the first two, by placing the words and fig- 
ures opposite each other : thus, 



First 
lesson. 



What it 
should con- 
tain. 



Second 
lesson. 



One - - 


- - 1 


Four - - 


- - 4 




Two - - 


- - 2 


Five - - 


- - 5 


Third 


Three - 


- - 3 


Six - - 


- - 6 


lesson. 


&c. 


&c. 


&c. 


&c. 





The Roman method of representing numbers 
should next be taught, making the fourth lesson • 
viz., 



208 



MATHEMATICAL SCIENCE. 



[bcok II. 



Fourth 
lesson. 



Roman 
method. 



One - 
Two - 
Three 
&c. 



I. 


Four 


II. 


Five 


III. 


Six 


&c. 


&c. 



IV. 
V. 

VI. 



First 
ten combi- 
nations : 



llow 

taught by 

things: 



§ 216. We come now to the first ten com- 
binations of numbers, which should be given in 
a separate lesson. In teaching them, we must, 

of course, have the aid of sensible objects. We 
teach them thus : 



One 


and 


one 


a it 


how 


many ? 






* 








One 


and 


two 


are 


how 


many ? 


* 




* * 








One 


and 


three 


arc 


how 


many ? 


* 




- 








&c. 




&CC. 




&C. a 





Hem in 
the abstract 



through all the combinations: after which, we 

pass to the abstract combinations, and ask, one 
and one are how many ? one and two. how 
many? one and three. &C. J after which we 
express the results in figures. 

We would then teach in the same manner, in 

second a separate lesson, the second ten combinations ; 

tions. then the third, fourth, fifth, sixth, seventh, eighth, 

ninth, and tenth. In the teaching of these com- 

Words u?ed. binations, only the words from one to twenty 

will have been used. We must then teach the 



CHAP. II.] ARITHMETIC TEXT-BOOKS. 



209 



combinations of which the results are expressed Further 
by the words from twenty to one hundred. tions. 



How 
they appear. 



§ 217. Having done this, in the way indi- Results. 
cated, the learner sees at a glance, the basis on 
w T hich the system of common numbers is con- 
structed. He distinguishes readily, the unit one 
from the unit ten, apprehends clearly how the 
second is derived from the first, and by com- 
paring them together, comprehends their mutual 
relation. 

Having sufficiently impressed on the mind of 
the learner, the important fact, that numbers are 
but expressions for one or more things of the 
same kind, the unit mark may be omitted in the unit mark 
combinations which follow. 



Same 
method in 
the other 

rules. 



§ 218. With the single difference of the omis- 
sion of the unit mark, the very same method 
should be used in teaching the one hundred 
combinations in subtraction, the one hundred 
and forty-four in multiplication, and the one 
hundred and forty-four in division. 

When the elementary combinations of the four 
ground rules are thus taught, the learner looks Results of 
back through a series of regular progression, in 
which every lesson forms an advancing step, 
and where all the ideas of each lesson have a 

14 



the method 



210 MATHEMATICAL SCIENCE. [BOOK II. 



mutual and intimate connection with each other. 

Are they Will not such a system of teaching train the 

mind to the habit of regarding each idea sepa- 

Thc rately — of tracing the connection between each 

power they . m 

C ive. new idea and those previously acquired — and ol 
comparing thoughts with each other? — and are 
not these among the guest ends to be attaii 

by instruction ? 



§ 2li). It has seemed to me of great im] 

ngm ance to use figures in the very first i is of 

should !><• . . . TT i i-Ti i 

used e«riy. arithmetic, unless this be done, the operations 

must all be conducted by means of sounds, and 
Rmmh, the pupil IS thus tatlght t6 regard sounds as the 

proper symbols of the arithmetical langus 

Oowe- This habit of mind, once firmly fixed, cannot 

wing words he easily eradicated; and when the figures are 
on> learned afterwards, they will not he regarded 
as the representatives of as many things as 
their names respectively import, hut as the rep- 
resentatives merely of familiar sounds which 
have been before learned. 

This would seem to account for the fact, 
about which, I believe, there is no difference of 
Oral opinion ; that a course of oral arithmetic, ex- 
tending over the whole subject, without the aid 
and use of figures, is but a poor preparation 
for operations on the s.ate. It may, it is true, 



CH A?. II.] ARITHMETIC TEXT-BOOKS. 



211 



sharpen and strengthen the mind, and give it what 

ii i i • • • i i ir raa >' do - 

development : but does it give it that language 

and those habits of thought, which turn it into what it 

r mi r does not do. 

the pathways of science ? The language of a 
science affords the tools by which the mind Language 

. . of arithmetic: 

pries into its mysteries and digs up its hidden 
treasures. The language of arithmetic is formed 
from the ten figures. Bv the aid of this lan- 
guage we measure the diameter of a spider's 
web, or the distance to the remotest planet 
which circles the heavens ; by its aid, we cal- 
culate the size of a grain of sand and the mag- 
nitude of the sun himself: should we then aban- 
don a language so potent, and attempt to teach its value, 
arithmetic in one which is unknown in the 
higher departments of the science ? 



Its uses. 



What 
it performs. 



§ 220. We next come to the question, how Fractions 
the subject of fractions should be presented in 
an elementary work. 

The simplest idea of a fraction comes from 
dividing the unit one into two equal parts. To 
ascertain if this idea is clearly apprehended, put 
the question, How T many halves are there in 
one ? The next question, and it is an import- 
ant one, is this.: How many halves are there in 
one and one-half? The next, How many halves 
in two? How many in two and a half? In 



implest 
idea. 



How 
impressed. 

Next 
question. 



212 MATH E MATICAL SCIENCE. [BOOK II 



three ? Three and a half? and so on to twelve. 
Result You will thus evolve all the halves from the 
units of the numbers from one to twelve, in- 
clusive. We stop here, because the multipli- 
cation table goes no further. These combina- 

First lesson, tions should be embraced in the first lesson on 
fractions. That lesson, therefore, will teach the 

its extent, relation between the unit 1 and the halves, and 
point out how the latter are obtained from the 
former. 

second § 221. The second lesson should be the first, 
es8on * reversed. The first question is, how man} 

GndM whole things are there in two halves? Sec- 
qoMUoos. ^^ j| ()W UVAU y whole things in four hah 

How many in eight? and so on to twenty-four 

halves, when we reach the extent of the division 

Extern of table. In this lesson you will have taught the 

e csson. p-j j Q p^ jjggjj f rom the fractions to the unit 

from which they are derived. 

fundamental § 222L You have thus taught the two fundft- 

pnnapio.: ^^^j p 1 i llc iplos of all the operations in frac- 
tions : viz. 
First, 1st. To deduce the fractional units from in- 

teger units ; and, 
rfceond. 2dly. To deduce integer units from fractional 
units 






CHAP. II.] ARITHMETIC TEXT-BOOKS. 213 



§ 223. The next lesson should explain the law Lessons 

explaining 

by which the thirds are derived from the units thirds. 
from 1 to 12 inclusive ; and the following lesson 
the manner of changing the thirds into integer 
units. 

The next two lessons should exhibit the same Fourths 

and other 

operations performed on the fourth, the next fractions. 
two on the fifth, and so on to include the 
twelfth. 



§ 224. This method of treating the subject of Advantages 

of the 

fractions has many advantages : method. 

1st. It points out, most distinctly, the relations 
between the unit 1 and the fractions which are First, 
derived from it. 

2d. It points out clearly the methods of pass- Second, 
ihg from the fractional to the integer units. 

3d. It teaches the pupil to handle and com- Third, 
bine the fractional units, as entire things. 

4th. It reviews the pupil, thoroughly, through Fourth, 
the multiplication and division tables. 

5th. It awakens and stimulates the faculties Fifth. 
of apprehension, comparison, and classification. 



§ 225. Besides the subjects already named, what 

the First Arithmetic should also contain the Arithmetic 

tables of denominate numbers, and collections slM ^ w 
of simple examples, to be worked on the slate, 



214 MATHEMATICAL SCIENCE. L BOOKH 

Examples, under the direction of the teacher. It is not 

how taught. 

supposed that the mind of the pupil is suffi- 
ciently matured at this stage of his progress to 
understand and work by rules. 

what § 22G. In the First Arithmetic, therefore, 

taught in tne pupil should be taught, 

the First j st rp^ l anguage Q f 6^^. 

Arithmetic. O D D 

2d. The four hundred and eighty-eight ele- 
second. mentary combinations, and the words by which 
they arc expressed ; 

Third. 8d. The main principles of Fractions ; 

Fourth. 4th. The tables of Denominate Numbers; and, 

Fifth. 5th. To perform, upon the slate, the element- 

ary operations in the four ground rule 



SECOND ARITIIMKTIf. 

Second § 227. This arithmetic occupies a largo space 

Arithmetic. . 

in the school education ot the country. .Many 
study it, who study no other. It should, there- 
What it fore, be complete in itself. It should also be 
eminently practical; but it cannot be made so 
either by giving it the name, or by multiplying 
the examples. 



should be. 



Practical § 228. The truly practical cannot be the ante- 

ipplication of 

principle, cedent, but must be the consequent of science. 



CHAP. II. J ARITHMETIC TEXT-BOOKS. 215 



Hence, that general arrangement of subjects Arrangement 

of subjects. 

demanded by science, and already explained, 
must be rigorously followed. 

But in the treatment of the subjects them- Reasons for 

i • • departures. 

selves, we are obliged, on account of the limited 
information of the learners, to adopt methods of 
teaching less general than we could desire. 



§ 229. We must here, again, begin with the Basis, 
unit one, and explain the general formation of 
the arithmetical language, and must also ad- 
here rigidly to the method of introducing new Method, 
principles or rules by means of sensible objects. 
This is most easily and successfully done either ** 

carried out. 

by an example or question, so constructed as to 
show the application of the principle or rule. 
Such questions or examples being used merely 
for the purpose of illustration, one or two will Few 

examples. 

answer the purpose much better than twenty : 

for, if a large number be employed, they are Reasons. 

regarded as examples for practice, and are lost 

sight of as illustrations. Besides, it confuses 

the mind to drag it through a long series of 

examples, before explaining the principles by 

which they are solved. One example, wrought one example 

under a rulo. 

under a principle or rule clearly apprehended, 
conveys to the mind more practical informa- 
tion; than a dozen wrought out as independent 



216 



MATHEMATICAL SCIENCE. [BOOK II. 



Principle, exercises. Let the principle precede the prac- 
Practice. tice, in all cases, as soon as the information 

acquired will permit. This is the golden rule 

both of art and morals. 



Its value 
in Addition 



subject* § 230. The Second Arithmetic should em- 

cmbraced. . r . 

brace all the subjects necessary to a lull view 
of the science of numbers ; and should contain 
an abundance of examples to illustrate their 
Reading: practical applications. The reading of muni 

so much (though not too much) dwelt upon, is 
an invaluable aid in all practical operations. 

By its aid, in addition, the eye runs up the 
columns and collects, in a moment, the sum of 
Subtraction : all the numbers. In subtraction, it glances at 
the figures, and the result is immediately sug- 
gested. In multiplication, also, the sight of the 
figures brings to mind the result, and it is 

reached and expressed by one word instead of 
five. In short division, likewise,, there is a cor- 
responding saving of time by reading the results 
of the operations instead of spelling them. The 
method of reading should, therefore, be con- 
stantly practised, and none other allowed. 



Multi- 
plication : 



Division. 



CHAP. II.] ARITHMETIC TEXT-BOOKS. 217 



THIRD ARITHMETIC. 

§ 231. We have now reached the place where Third 

... . . . ,_. Arithmetic 

arithmetic may be taught as a science, lhe 
pupil, before entering on the subject as treated Preparation 
here, should be able to perform, at least mechan- 
ically, the operations of the five ground rules. 

Arithmetic is now to be looked at from an 
entirely different point of view. The great view of u. 
principles of generalization are now to be ex- 
plained and applied. 

Primarily, the general language of figures What 

i i ii -I • r i i3 taught 

must be taught, and the striking fact must then primarily. 

be explained, that the construction of all integer 

numbers involves but a single principle, viz. 

the law of change in passing from one unit to General law 

another. The basis of all subsequent operations 

will thus have been laid. 

§ 232. Taking advantage of this general law 
which controls the formation of numbers, we controls 

. . , , . . p formation of 

bring all the operations "of reduction under one numbers 
single principle, viz. this law of change in the 
unities. 

Passing to addition, we are equally surprised its value 
and delighted to find the same principle con- 
trolling all its operations, and that integer num- 
bers of all kinds, whether simple or denominate, 
may be added under a single rule. 



218 MATHEMATICAL SCIENCE. [bOOKII. 

Advantages This view opens to the mind of the pupil a 

of knowing a • i r ^ ^ r l 1 t i r> -ii 

general law. wide field ol thought. It is the first illustra- 
tion of the great advantage which arises from 
looking into the laws by which numbers are 

subtraction, constructed. In subtraction, also, the same 
principle finds a similar application, and a sim- 
ple rule containing but a few words is found 
applicable to all the classes of integer numbers. 

In multiplication and division, the same stri- 
king results flow from the same cause; and 
General thus this simple principle, viz. the law of change 

law of iiuin- . r - r i t 

ben: in passing jroni one unit of value to another, is 
the hey to all the operations in the four ground 
rules, whether performed on simple or denomi- 
nate numbers. Thus, all the elementary opera* 
Controls tions of arithmetic are linked to a single prin- 
tion. L ciple, an( l that one a mere principle of arith- 
metical language. Who can calculate the la- 
bor, intellectual and mechanical, which may 
be saved by a right application of this lumin- 
ous principle ? 

Design § 233. It should be the design of a higher 

°arithmeUcT arithmetic to expand the mind of the learnei 
over the whole science of numbers ; to illus- 
trate the most important applications, and to 
make manifest the connection between the sci- 
ence and the art. 



CHAP. II. J ARITHMETIC TEXT-BOOKS. 219 

It will not answer these objects if the methods its 

. . requisites. 

of treating the subject are the same as m the 
elementary works, where science has to com- 
promise with a want of intelligence. An ele- 
mentary is not made a higher arithmetic, by Must have 

, r ' •!/••• • ••! a distinctive 

merely transferring its definitions, its principles, character. 
and its rules into a larger book, in the same 
order and connection, and arranging under them 
an apparently new 7 set of examples, though in fact 
constructed on precisely the same principles. 

§ 234. In the four ground rules, particularly constmo 

... i*i tion °* exau * 

(where, in the elementary works, simple exam- pies in the 

pies must necessarily be given, because here J^^ 
they are used both for illustration and practice), 
the examples should take a wide range, and be 
so selected and combined as to show thei r com- 
mon dependence on the same principle. 

§ 235. It being the leading design of si series Desigu 
of arithmetics to explain %nd JIustrate the sci- 
ence and art of numbers, great care should be 
taken to treat all the subjects, as far as their 
different natures will permit, according to the 
same general methods. In passing from one 
book to another, every subject which has been subject* 
fully and satisfactorily treated in the one, should fe ° r Jj when 
be transferred to the other with the fewest pos- fu,ly treated 



220 MATHEMATICAL SC/ENCE. [b JOK IT. 

How com- sible alterations ; so that a pupil shall not have 

mon subjects . 

maybe to learn under a new dress that which he has 
already fully acquired. They who have studied 
the elementary work should, in the higher one, 
either omit the common subjects or pass thorn 
over rapidly in review. 

The more enlarged and comprehensive views 

lUaaons. which should l>e given in the higher work will 

thus l)o acquired with the least possible labor, and 

the connection of the series clearly pointed out. 

This use of those subjects, which have been 

fully treated in the elementary work, is greatly 

preferable to the method of attempting to teach 

Additional every thing anew : for there must necessarily be 

stated. much that is common : and that which teaches 

no now principle, or indi 'atea no now method of 

application, should be precisely the same in the 

higher work as in that which precedes it. 

§ 236. To vary the examples, in form, without 

changing in the least the principles on which 

a contrary they are worked, and to arrange a thousand such 

metnod leads . 

to confusion: collections under the same set of rules and sub- 
ject to the same laws of solution, may gire a 
little more mechanical facility in the use of 
figures, but will add nothing to the stores of 
arithmetical knowledge. Besides, it deludes the 
learner with the hope of advancement, and when 



CHAP. II.} ARITHMETIC CONCLUSION. 221 



he reaches the end of his higher arithmetic, he it misleads 

the pupil : 

finds, to his amazement, that he has been con- 
ducted by the same guides over the same ground 

through a winding and devious way, made it com- 
plicates th» 
strange by fantastic drapery : whereas, if what subject. 

was new had been classed by itself, and what 
was known clothed in its familiar dress, the sub- 
ject would have been presented in an entirely 
different and brighter light. 



CONCLUDING RE M ARKS. 

We have thus completed a full analysis of the conclusion 
language of figures, and of the construction of 
numbers. 

We have traced from the unit one, all the what 
numbers of arithmetic, whether integer or frac- d one. 
tional, whether simple or denominate. We have 
developed the laws by which they are derived Laws, 
from this common source, and perceived the 
connections of each class with all the others. 

We have examined that concise and beautiful Analysis 
language, by means of which numbers are made gua ge. 
available in rendering the results of science 
practically useful ; and we have also considered Methods 

°** teaching 

the best methods of teaching this great subject indicated. 

— the foundation of all mathematical science — import- 
ance of ths 
and the first among the useful arts. subject. 



CHAP. III.] GEOMETRY. 223 



CHAPTER III. 

GEOMETRY DEFINED — THINGS CF WHICH IT TREATS — COMPARISON AND PROP- 
ERTIES OF FIGURES — DEMONSTRATION — PROPORTION — SUGGESTIONS FOR 
TEACHING. 

GEOMETRY. 

§ 237. Geometry treats of space, and com- Geometry, 
pares portions of space with each other, for the 
purpose of pointing out their properties and mu- 
tual relations. The science consists in the de- ita science, 
veiopment of all the laws relating to space, and 
is made up of the processes and rules, by means 
of which portions of space can be best compared 
with each other. The truths of Geometry are a Its trutns - 
series of dependent propositions, and may be di- 0f three 
vided into three classes : 

1st. Truths implied in the definitions, viz. that 1st Those 

... . implied in 

things do exist, or may exist, corresponding to tnedefim- 
the words defined. For example : when we say, faona * 
" A quadrilateral is a rectilinear figure having four 
sides," we imply the existence of such a figure. 

2d. Self-evident, or intuitive truths, embodied **■ Axiotns » 
in the axioms ; and, 

3d. Truths inferred from the definitions a™* 3d. Demon- 



224 MATHEMATICAL SCIENCE. [BOOK II, 

etrative axioms, called Demonstrative Truths. We say 
that a truth or proposition is proved or demon- 

When de- 
monstrated, strated, when, by a course of reasoning, it is 

shown to be included under some other truth or 
proposition, previously known, and from which 
is said to follow; hence, 
Demoustra- A Demonstration is a scries of logical argu- 
ments, brought to a conclusion, in which the 
major premises are definitions, axioms, or prop- 
ositions already established. 

Bnttfectoof §238. Before we can understand the proofi 
Geometry. , , 

or demonstrations ol beometry, we must under- 
stand what that is to which demonstration is 
applicable: hence, the first thing necessary is 
to form a clear conception of space, the subject 
of all geometrical reasoning. 1 
Nomesof The next step is to give names to particular 
form*. f orms or limited portions of space, and to define 
these names accurately. The definitions of these 
names are the definitions of Geometry, and the 
portions of space corresponding to them are 
Figures, called Figures, or Geometrical Magnitudes ; ol 
Three kinds, which there are three general classes : 
rirst# 1st. Lines; 

second. 2d. Surfaces; 
Third. 3d. Solids. 

* Sections 81 to 85. 



CHAP. III.] GEOMETRY. 225 

§ 239. Lines embrace only one dimension of Lines, 
space, viz. length, without breadth or thickness. 
The extremities, or limits of a line, are called 
points. 

There are two general classes of lines — straight Two classes: 

.. , , ,. . '11- • Straight and 

lines and curved lines. A straight line is one curved, 
which lies in the same direction between any 
two of its points ; and a curved line is one which 
constantly changes its direction at every point. 
There is but one kind of straight line, and that is one kind of 

. straight line 

fully characterized by the definition. From the 
definition we may infer the following axiom : " A 
straight line is the shortest distance between two 
points." There are many kinds of curves, of many of 
which the circumference of the circle is the sim- 
plest and the most easily described. 

§ 240. Surfaces embrace two dimensions of surfaces: 
space, viz. length and breadth, but not thickness. 
Surfaces, like lines, are also divided into two rianeand 
general classes, viz. plane surfaces and curved 
surfaces. 

A plane surface is that with which a straight a plane 
line, any how placed, and having two points 
common with the surface, will coincide through- 
out its entire extent. Such a surface is per- 
fectly even, and is commonly designated by the Perfectly 

term " A plane," A large class of the figures 

15 



226 MATHEMATICAL SCIENCE. [bOCKII. 



Dane rig- of Geometry are but portions of a plane, and all 
such are called plane figures. 

§ 241. A portion of a plane, bounded by three 
a triangle, straight lines, is called a triangle, and this is the 

the most sim- 
ple figure, simplest of the plane figures. There are several 

kinds of triangles, differing from each other, 

however, only in the relative values of their 

sides and angles. For example: when the sides 

are all equal to each other, the triangle is called 

Kinds of in- equilateral ; when two of the side- are equal, it 

is called isosceles : and scalene, when the three 
sides are all unequal. If one of the angles is a 

right angle, the triangle Is called a right-angled 
triangle. 

§ 24J. The next simplest class of plane figures 
comprises all those which are hounded by four 

Quadrikiter- straight lilies and are called quadrilaterals. 

There are several varieties of this clac 

1st species, 1st. The mere quadrilateral, which has no 
mark, except that of having four sides ; 

2d species. 2d. The trapezoid, which has two sides par- 
allel and two not parallel ; 

3d species. 3d. The parallelogram, which has its opposite 
sides parallel and its angles oblique ; 

«h species. 4th. The rectangle, which has all its angles 
right angles and its opposite sides parallel ; and. 






CHAP. III. J GEOMETRY. 227 

5th. The square, which has its four sides equal 5th species, 
to each other, each to each, and its four angles 
right angles. 

§243. Plane figures, bounded by straight lines, other Poiy- 
Aaving a number of sides greater than four, take 
names corresponding to the number of sides, viz. 
Pentagons, Hexagons, Heptagons, &c. 

§ 244. A portion of a plane bounded by a circles: 
curved line, all the points of which are equally 
distant from a certain point within called the 
centre, is called a circle, and the bounding line 
is called the circumference. This is the only the circum 
curve usually treated of in Elementary Geometry. 

§ 245. A curved surface, like a plane, em- curved sur- 
faces : 
braces the two dimensions of length and breadth. 

It is not even, like the plane, throughout its whole 

extent, and therefore a straight line may have their proper- 

two points in common, and yet not coincide with 

it. The surface of the cone, of the sphere, and 

cylinder, are the curved surfaces treated of in 

Elementary Geometry. 



ties. 



§ 246. A solid is a portion of space, combi- solids, 
ning the three dimensions of length, breadth, and 
thickness, Solids are divided into three classes: Three classes, 



228 MATHEMATICAL SCIENCE. [boOKIJ. 

1st cia&j. 1st. Those bounded by planes ; 
2<j cia»s. 2d. Those bounded by plane and curved sur- 
faces ; and., 
3d class. 3d. Those bounded only by curved surfaces, 

what figures The first class embraces the pyramid and 
dan. prism with their several varieties; the second 
class embraces the cylinder and cone ; and the 
third class the sphere, together with others not 
generally treated of in Elementary Geometry. 

Magnitude! § 217. We have nOW named all the geomet- 
rieal magnitudes treated of in elementary G 

whatthey ometry. They are merely limited portions 
space, and do not, necessarily, involve the i< 

Atpbere. 6f matter. A sphere, for example, fulfils all the 
conditions imposed by its defini dons, without any 
reference to what may be within the space en- 

Need not be closed bv its surfaee. That space may be OC« 
cupied by lead. iron, or air. or may he a vaeinnn. 
without at all changing the nature of the sphere, 
as a geometrical magnitude. 

It should be observed that the boundary or 
Boundaries limit of a geometrical magnitude, is another geo- 
ofsohd9 ' metrical magnitude, having one dimension 1. 

For example : the boundary or limit of a solid. 
Examples, which has three dir ensions, is always a surfa 

which has but two the limits or boundaries ol 



CHAP. III.] GEOMETRY. 



220 



all surfaces are lines, straight or curved ; and the 
extremities or limits of lines are points. 

§ 248. We have now named and shown the subject* 

named. 

nature of the things which are the subjects of 
Elementary Geometry. The science of Ge- s <^ceof 

Geometry. 

ometrv is a collection of those connected pro- 
cesses by which we determine the measures, 
properties, and relations of these magnitudes. 



COMPARISON OF FIGURES WITH UNITS OF MEASURE. 

§ 249. We have seen that the term measure Measure. 
implies a comparison of the thing measured with 
some known thing of the same kind, regarded 
as a standard ; and that such standard is called 
the unit of measure.* The unit of measure for unitofmea§ 

ure 

lines must, therefore, be a line of a known length : For Line*, 

a foot, a yard, a rod, a mile, or any other known 

unit. For surfaces, it is a square constructed For surfaces, 

on the linear unit as a side : that is, a square a square, 

foot, a square yard, a square rod. a square mile ; 

iftat is, a square described on any known unit 

of length. 

The unit of measure, for solidity, is a solid, ForSoiids 
and therefore has three dimensions. It is a cube a Cube. 

* Section 94. 



2,'JO M A T II E MATICAL SCI E N C E . [BOOK II. 



constructed on a linear unit as an edge, or on 

the superficial unit as a base. It is, therefore, 

a cubic foot, a cubic yard, a cubic rod, &c. 

itowvtta Hence, there are three units of measure, each 

differing in kind from the other two, viz. a known 

a Lino, length for the measurement of lines; a known 

A Square, square for the measurement of surfaces; and a 

a cub.-, known cube for the measurement of solids. The 

Uontonto: measure or contents of any magnitude, bdo 

bow Mo*- ing to either class, is ascertained by finding how 

many times that magnitude contains its unit of 

measure. 

§ 250. There is yet another class of magni- 
tudes with which Geometry is conversant, called 

Angica: Angles. They are not, however, elementary 
magnitudes, but arise from the relative positions 
Their unit, t>f those already described. The unit of this 
class is the right angle ; and with this as B stand- 
ard, all other angles are compared 



§ 251. We have dwelt with much detail on 

the unit of measure, because it furnishes the 

importance only basis of estimating quantity. The con- 

of the unit of 

measure: ception of number and space merely opens to 
the intellectual vision an unmeasured field of 
investigation and thought, as the ascent to the 
summit of a mountain presents to the eve a 



CHAP. III.] GEOMETRY. 231 



wide and unsurveyed horizon. To ascertain the space indefl- 

..,.., . r it r \ nite without 

height 01 the point 01 view, the diameter oi the it: 

surrounding circular area and the distance to 

any point which may be seen, some standard or 

unity must be known, and its value distinctly 

apprehended. So, also, number and space, which 

at first fill tne mind with vague and indefinite and always 

measured 

conceptions, are to be finally measured by units by it. 
of ascertained value. 



§ 252. It is found, on careful analysis, that Every num- 
every number maybe referred to the unit one, ^0^^ 
as a standard, and when the signification of the lhe unit one " 
term one is clearly apprehended, that any num- 
ber, whether large or small, whether integer or 
fractional, may be deduced from the standard by 
an easy and known process. 

In space, also, which is indefinite in extent, space: 
and exactly similar in all its parts, the faculties 
of the mind have established ideal boundaries, its ideal 
These boundaries give rise to the geometrical vm anea * 
magnitudes, each of which has its own unit of 
measure ; and by these simple contrivances, we 
measure space, even to the stars, as with a yard- 
stick. 

§ 253. We have, thus far, not alluded to the 
difficulty of determining the exact length of that 



232 MATHEMATICAL SCIENCE. L B00K H 

conception which we regard as a standard. We are pre- 

of the unit of . . 

measure: sented with a given length, and told that it is 
called a foot or a yard, and this being usually 
done at a period of life when the mind is satis- 
fied with mere facts, we adopt the conception 
At first, a of a distance corresponding to a Dame, and then 
mere oi !!! prei bv multiplying and dividing that distance we 
are enabled to apprehend other distances. Hut 
this by n<> means answers the inquiry, What is 
the standard t<>r measurement ? 
nowdeter- Under the supposition that the laws of phys- 
minCiL ical nature operate uniformly, the unit of me 

ure in England and the United States has been 
fixed by ascertaining the length of a pendulum 
which will vibrate seconds, and to this length 

the Imperial yard, which we have also adopted 

as a standard, is referred. Hence, the unit <>t 

whatitis. measure is referred to a natural standard, viz. to 

the distance between the axis of suspension and 

the centre of oscillation of a pendulum which 

shall vibrate seconds in vacuo, in London, at the 

level of the sea. This distance is declared to 

its lenprth. be 30.1393 imperial inches; that is, 3 imperial 

feet and 3.1393 inches. Thus, the determina- 

LMfflcuitics lion of the unit of length demands the applica- 

tosttT tion of the most abstruse science, combined with 

accurate observation and delicate experiment. 

Could this distance, or unit, have been exactly 



CHAP. III.] GEOMETRY. 233 

ascertained before the measures of the world 
were fixed, and in general use, it would have what should 
afforded a standard at once certain and conve- called one. 
nient, and all distances would then have been other num- 

. . . r . . ,. bers derived 

expressed in numbers arising from its multiph- from it. 

cation or exact division. But as the measures 

of the world (and consequently their units) were why it is not 

fixed antecedently to the determination of this 

distance, it was expressed in measures already 

known ; and hence, instead of being represented 

by 1, which had already been appropriated to what now 

represents it. 

the foot, it was expressed in terms of the foot, 
viz. 39.1393 inches, and this is now the standard 
to which all units of measure are referred. 



§ 254. The unit of measure is not only im- unit of mea* 

ure the basis 

portant as affording a basis for all measurement, of the unite 
but is also the element from which we deduce 
the unit of weight. The weight of 27.7015 cubic 
inches of distilled water is taken as the standard, 
weighing exactly one pound avoirdupois, and this 
quantity of water is determined from the unit 
of length; that is, the determination of it reaches what it is. 
back to the length of a pendulum which will 
vibrate seconds in the latitude of London. 

§ 255. Two geometrical figures are said to be Equivalent 
equivalent, when they contain the same unit o r gure8 ' 



234 M ATHE M ATICAL SCIENCE. [BOOK IT. 

measure an equal number of times. Two figures 

Equal fig- are said to be equal when they can be so applied 

to each other as to coincide throughout their 

Equivalency: whole extent. Hence, equivalency refers to 
Equality. measure, and equality to coincidence. Indeed, 
coincidence is the only test of geometrical equal- 
ity. All equal figures are of course equivalent, 

Th.-ir (litT.r- though equivalent figures are by no menus equal. 
Equality is equivalency, with the further mark 
of coincidence. 



r I o P B B T IKS OF f I a B B S. 
property or § 256, A property of B figure IB ■ mark com- 

li'-ruri'S. 

mon to all figures of the same class. For exam- 

u.uuiriiater- pj ( > ; \( the r l;iss he " (Quadrilateral," there are two 

J ils. 

very obvious properties, common to all quadri- 
laterals, besides the one which characterizes 
the figure, and by which its name is defined, 
viz. that it has four angles, and that it may 
be divided into two triangles. If the class be 

p.iraiieio- " Parallelogram/' there are several properties 
common to all parallelograms, and which are 
subjects of proof ; such as, that the opposite 
sides and angles are equal ; the diagonals divide 
each other into equal parts, Szc. If the class be 

Triangle: "Triangle," there are many properties common 
to all triangles, besides the characteristic that 



CHAP. III.] GEOMETRY 235 

they have three sides. If the class be a par- Equilateral, 
ticular kind of triangle, such as the equilateral, isosceles, 
isosceles, or right-angled triangle, then each class Right-angled, 
has particular properties, common to every indi- 
vidual of the class, but not common to the other 

classes. It is important, however, to remark, Every prop- 
erty which 
that every property which belongs to * triangle," belong to a 

. . .,. . genus will be 

regarded as a genus, will appertain to every comm onto 
species or class of triangle: and universally. ever >' 9 P e " 

r o J v ' cies: 

every property which belongs to a genus will 
belong to every species under it ; and every 
property which belongs to a species will be- 
long to every class or subspecies under it; and alsolo every 
even* property which belongs to one of a sub- sub8 P ecies » 

* * •» o an( j t e v er y 

species or class will be common to every indi- individual - 
vidual of the class. For example : " the square Examples, 
on the hypothenuse of a right-angled triangle is 
equivalent to the sum of the squares described 
on the other two sides,'' is a proposition equallv 
true of every right-angled triangle: and "every 
straight line perpendicular to a chord, at the circle, 
middle point, will pass through the centre/' is 
equally true of all circles. 



MARKS OF WHAT MAY BE PROVED. 

§ 257. The characteristic properties of every character!* 
geometrical figure (that is, those properties with- tic i ^° per " 



236 M A THE M A T I C A L SCIE N C E . L B00K !I * 

out which the figures could not exist), are given 
in the definitions. How are we to arrive at all 
the other properties of these figures? The 
propositions implied in the definitions, viz. that 
Marks: things corresponding to the words defined do or 
may exist with the properties named ; and the 
Of what may self-evident propositions or axioms, contain the 

be proved. 

only marks of what can be proved ; and l>y a 
Howex- s kj]j' u i combination of these marks we arc able 

tended. 

to discover and prove all that is discovered and 
proved in ( reometry. 

Definitions and axioms, and propositions de- 

M:,J,,r duced from them, are major premises in each 

The Rtoooe: new demonstration ; and the science IS made up 

ooiMMa. °f tnr processes employed for bringing unfore- 
seen cases under these known truths ; or, ID syl- 
logistic language, for proving the minors m-< 
sary to complete the syllogisms. The marks 

being so few, and the inductions which furnish 
them so obvious and familiar, there would seem 
to be very little difficulty in the deductive pro- 
cesses which follow. The connecting together 
of several of these marks constitutes Deductions, 
Geometry, or Trains of Reasoning; and hence. Geometry 
is a Deductive Science. 



i Deductive 
Science. 



CHAP. III.] 



GEOMETRY. 



237 



to be proved. 



Enunciation. 



DEMONSTRATION. 

§ 258. As a first example, let us take the first 
proposition in Legendre's Geometry : 

"If a straight line meet another straight line, Proposition 
the sum of the two adjacent angles will be equal 
to two right angles" 

Let the straight line DC 
meet the straight line AB 
at the point C, then will the 
angle ACD plus the angle 
DCB be equal to two right AC B 

angles. 

To prove this proposition, we need the defini- 
tion of a right angle, viz. : 

When a straight line AB B 

meets another straight line 
CD. so as to make the ad- 
jacent angles BAC and 
BAD equal to each other, 
each of those angles is called a right angle, and 
she line AB is said to be perpendicular to CD. 

We shall also need the 2d, 3d, and 4th axioms, *=*■* 
for inferring equality,* viz. : 

2. Things which are equal to the same thing second, 
are equal to each other. 



Things 

necessary to 

prove it. 



D Definition* 



f Section 102. 



238 



MATHEMATICAL SCIENCE. 



[book II 



Third. 



Fourth. 



Proof: 




Continued : 



Conclusion. 



Ita bases. 



First. 



3. A whole is equal to the sum of all it* 
parts. 

4. If equals be added to equals, the sums 
will be equal. 

Now before these formulas or tests can be ap- 
plied, it is necessary to sup- k n 
pose a straight line CE to be 
drawn perpendicular to AV> 
at the point C: then by the 
definition of a right angle, A B 
the angle ACE will be equal to the angle ECB, 

By axiom 3rd, we h;i\ 

ACD equal to ACE pins BCD: to each of 
these equals add DCB; and by the ith axiom 

we shall have, 

ACD plus DCB equal to AX3E plus ECD plus 

DCB; but by axiom 3rd, 
ECD plus DCB equals ECB: therefore bj 

axiom 2d, 

ACD plus DCB equals ACE plus ECB. 

But by the definition of a right angle, 

ACE plus ECB equals two right angles : there- 
fore, by the 2d axiom, 

ACD plus DCB equals two right angles. 

It will be seen that the conclusiveness of the 
proof results, 

1st. From the definition, that ACE and ECB 
are equal to each other, and each is called a 



CHAP. III.] fiEO M ETRY. 239 

right-angle : consequently, their sum is equal to 
two right angles ; and, 

2dly. In showing, by means of the axioms, that Secoc*. 
ACD plus DCB equals ACE plus ECB; and 
then inferring from axiom 2d, that, ACD plus 
DCB equals two right angles. 

§ 259. The difficulty in the geometrical rea- Acuities in 

the demon- 

soning consists mainly in showing that the prop- stations. 
osition to be proved contains the marks which 
prove it. To accomplish this, it is frequently 
necessary to draw many auxiliary lines, forming Auxiliaries 

necessary. 

new figures and angles, which can be shown to 
possess marks of these marks, and which thus 
become connecting links between the known connecting 
and the unknown truths. Indeed, most of the 
skill and ingenuity exhibited in the geometrical 
processes are employed in the use of these auxil- 
iary means. The example above affords an illus- 
tration. We were unable to show that the sum How used, 
of the two angles possessed the mark of being 
equal to two right angles, until we had drawn a 
perpendicular, or supposed one drawn, at the 
point where the given lines intersect. That be- 
ing done, the two right angles ACE and ECB <>-*"*■* 
were formed, which enabled us to compare the 
sum of the angle ACD and DCB with two right 
angles, and thus we proved the proposition. 



240 



MATHEMATICAL SCIENCE. [BOOK II. 



Diagram. 




Principles 
necessary. 



Proposition. § 260. As a second example, let us take the 

following proposition : 
Enunciation. If two straight lines meet each other, the op- 
posite or vertical angles icill be equal. 

Let the straight line A 
AB meet the straight line 
ED at the point C : then 
will the angle ACD be 
equal to the opposite an- 
gle ECB; and the an<_ r le ACE equal to the an- 
gle DCB. 

To prove this proposition, we need the last 
proposition, and also the fc Jd and 5th axioms, viz. : 

"If a Straight line meet another straight line. 

the sum of the two adjacent angles will be equal 
to two right angles." 

"Things which are equal to the same thing 
are equal to each other." 

"If equals be taken from equals, the remain- 
ders will be equal." 

Now r , since the straight line AC meets the 
straight line ED at the point C, we have, 

ACD plus ACE equal to two right ^angles. 

And since the straight line DC meets the 
straight line AB, we have, 

ACD plus DCB equal to two right angles : 
hence, by the second axiom. 

ACD plus ACE equals ACD plus DCB : ta- 



Axioms. 



Proof. 






CHAP. III.] GEOMETRY. 241 

king from each the common angle ACD, we conclusion, 
know from the fifth axiom that the remain- 
ders will be equal ; that is, the angle ACE 
equal to the opposite or vertical angle DCB. 

§ 261. The two demonstrations given above 
combine all the processes of proof employed in D emonstra- 
every demonstration of the same class. When tions generaK 
any new truth is to be proved, the known tests 
of truth are gradually extended to auxiliary Useof auxil . 
quantities having a more intimate connection mry quanti " 

n & ties. 

with such new truth than existed between it and 
the known tests, until finally, the known tests, 
through a series of links, become applicable to 
the final truth to be established : the interme- 
diate processes, as it were, bridging over the 
space between the known tests and the final 
truth to be proved. 

§ 262. There are two classes of demonstra- Direct dem 
tions, quite different from each other, in some 
respects, although the same processes of argu- 
mentation are employed in both, and although 
the conclusions in both are subjected to the 
lame logical tests. They are called Direct, or „ 

° J Negative, 

Positive Demonstration, and Negative Demon- or 

Reductio ad 

6tration. or the Reductio ad Absurdum. Absurdum. 



16 



242 MATHEMATICAL SCIENCE. [bOOKII. 

Difference. § 263. The main differences in the two 
methods are these : The method of direct demon- 
Direct Bem- stration rests its arguments on known and ad- 

onstration. 

mitted truths, and shows by logical processes 

that the proposition can be brought under some 

previous definition, axiom, or proposition : while 

Negative the negative demonstration rests its arguments 

Demonstra- 
tion, on an hypothesis, combines this with known pro- 
positions, and deduces a conclusion by processes 

Conclusion: strictly logical. Now if the conclusion so de- 
duced agrees with any known truth, we inter 

with what that the j lvpot ] lesis ( W l uc h WM t h e on i v ii n k in 

compared. * l > 

the chain not previously known), was true; but 
if the conclusion be excluded from the truths 
previously established ; that is, if it be opposed 
to any one of them, then it follows thai the hy- 
pothesis, being contradictory to such truth, must 
Determines ^ e f a ] se j n the negative demonstration, theiv- 

whether the 

hypothesis is fore, the conclusion is compared with the truths 

true or false. 

known antecedently to the proposition in ques- 
tion : if it agrees with any one of them, the hy- 
pothesis is correct ; if it disagrees with any one 
of them, the hypothesis is false. 



proof by § 264. We will give for an illustration of this 

Negative 

Demonstra- method, Proposition XVII. of the First Book of 
Legendre : " When two right-angled triangles 
have the hypothenuse and a side of the one equal 






CI1 AP. III. J GEOMETRY. 243 



to the hypothenuse and a side of the other, each Enunciation, 
to each, the remaining parts will be equal, each to 
each, and the triangles themselves will be equal."' 

In the two right-angled triangles BAC and 
EDF (see next figure), let the hypothenuse AC Enunciation 
be equal to DF, the side BA to the side ED : hy th 
then will the side BC be equal to EF, the angle 
A to the angle D, and the angle C to the angle F. 
To prove this proposition, w r e need the follow- 
ing, which have been before proved ; viz. : 

Prop. X. (of Legendre). "When two triangles previous 
have the three sides of the one equal to the three truths Dece * 

* sary. 

sides of the other, each to each, the three an- 
gles will also be equal, each to each, and the 
triangles themselves will be equal." 

Prop. V. " When two triangles have two Proposition 
sides and the included angle of the one, equal 
to tw r o sides and the included angle of the other, 
each to each, the two triangles will be equal.'*' 

Axiom I. " Things w r hich are equal to the Axioms, 
same thing, are equal to each other." 

Axiom X. (of Legendre). "All right angles 
are equal to each other." 

Prop. XV. " If from a point without a straight Proposition, 
line, a perpendicular be let fall on the line, and 
oblique lines be drawn to different points, 

1st. "The perpendicular frill be shorter than 
any oblique line ; 



244 



MATHEMATICAL SCIENCE. [BOOK II. 




Construction 
of the figure. 



2d. "Of two oblique lines, drawn at pleasure, 
that which is farther from the perpendicular will 
be the longer/' 
Now the two sides BC and 
Deginning cf EF are either equal or un- 

the demon- 
stration, equal. If they are equal, 

then by Prop. X. the remain- 
ing parts of the two trian- c G 
gles are also equal, and the triangles themselves 
are equal. If the two sides are unequal, 0116 of 
them must be greater than the other: supp 
BC to be the greater. 

On the greater side BC take a part BG, equal 
to EF, and draw A( r. Then, in the two trian- 
gles BAG and DEF the angle B is equal to the 
angle E, by axiom X (Legendre), both being 
right angles. The side AB is equal to the side 
DE, and by hypothesis the side BG is equal to the 
side EF: then it follows from Prop. V. that the 
side AG is equal to the side DF. But the side 
DF is equal to the side AC : hence, by axiom I, 
the side AG is equal to AC. But the line AG 
cannot be equal to the line AC, having been 
shown to be less than it by Prop. XV. : hence, 
Conclusion, the conclusion contradicts a known truth, and is 
therefore false ; consequently, the supposition (on 
which the conclusion rests), that BC and EF are 
unequal, is also false ; therefore, they are equal 



Demonstra- 
tion. 






CHAP. III.] GEOMETRY. 245 

§ 265. It is often supposed, though erroneous- Negative 
ly, that the Negative Demonstration, or the dem- tion . 
onstration involving the "reductio ad absurdum/' 
is less conclusive and satisfactory than direct or conclusive, 
positive demonstration. This impression is sim- 
ply the result of a want of proper analysis. For 
example : in the demonstration just given, it was Reasons. 
proved that the two sides BC and EF cannot 
be unequal; for, such a supposition, in a logi- 
cal argumentation, resulted in a conclusion di- conclusion 

. . , . , .. corresponds 

rectly opposed to a known truth; and as equality t0? oris op _ 
and inequality are the only general conditions P osedto 

^ J J ° known truth, 

of relation between two quantities, it follows 
that if they do not fulfil the one, they must the 
other. In both kinds of demonstration, the 
premises and conclusion agree ; that is, they are Agreement, 
both true, or both false ; and the reasoning or 
argument in both is supposed to be strictly logi- 
cal. 

In the direct demonstration, the premises are 
known, being antecedent truths ; and hence, 
the conclusion is true. In the negative demon- Differences in 
stration, one element is assumed, and the con- , e *° 

kinds. 

elusion is then compared with truths previously 
established. If the conclusion is found to agree 
with any one of these, we infer that the hy- when the 
pothesis or assumed element is true; if it con- hyp °^ sis,a 
tradicts any one of these truths, we infer that 



246 



MATHEMATICAL SCIENCE. 



[book II 



When false, the assumed element is false, and hence that its 
opposite is true. 



Measured : 
its significa- 



§ 266. Having explained the meaning of the 
term measured, as applied to a geometrical mag- 
nitude, viz. that it implies the comparison of a 
magnitude with its unit of measure ; and having 
also explained the signification of the word Prop- 
General erty, and the processes of reasoning by which, 

Remarks. 

in all figures, properties not before noticed are 
inferred from those that are known ; we shall 
now add a few remarks on the relations of the 
geometrical figures, and the methods of compar- 
ing them with each other. 



Proportion . 



PKOPORTION OF FIGURES. 

§ 267. Proportion is the relation which one 
geometrical magnitude bears to another of the 
same kind, with respect to its being greater or 
less. The two magnitudes so compared are called 
terms, and the measure of the proportion is the 
quotient which arises from dividing the second 
term by the first, and is called their Ratio. Only 
quantities of the same kind can be compared 
the same together, and it follows from the nature of the 

kind com- 
pared, relation that the quotient or ratio of any two 

terms will be an abstract number, whether the 

terms themselves be abstract or concrete. 



Its measure. 

Ratio. 
Quantities of 



CHAP. III.] GEOMETRY. 247 



§ 268. The term Proportion is defined by most Proportion: 
authors, " An equality of ratios between four 
numbers or quantities, compared together two 
and two." A proportion certainly arises from 
such a comparison : thus, if 

B D . _ . 

-T- = 7? ; then, Example. 

A O 
A : B : : C : D 

is a proportion. 

But if we have two quantities A and B, which True defini- 
tion, 
may change their values, and are, at the same 

time, so connected together that one of them 

shall increase or decrease just as many times as 

the other, their ratio will not be altered by such 

changes ; and the two quantities are then said Tw °P ro P° r ; 

to be in proportion, or proportional. ties - 

Thus, if A be increased three times and B 

three times, then, 

3BA 
3A~B ; 

that is, 3 A and 3 B bear to each other the same 
proportion as A and B. Science needed a gen- Term need- 

ed 

eral term to express this relation between two 
quantities which change their values, without 
altering their quotient, and the term "propor- 
tional," or "in proportion," is employed for that How used, 
purpose. 



248 MATHEMATICAL SCIENCE. [bOOKII, 

Reasons for As some apology for the modification of the 

modification. . r . . r . i • i i_ 1 i 

definition of proportion, which has been so long 

accepted, it may be proper to state that the term 

has been used by the best authors in the exact 

use of the sense here attributed to it. In the definition of 

term. 

the second law of motion, we have, ''Motion, 
or change of motion, is proportional to the foi 
impressed;" 41 and again, "The inertia of a body 
is proportioned to its weight."! Similar exam- 
ples may be multiplied to any extent. Indeed, 
Symbol used there is a symbol or character to express the 

to represent 

proportion, relation between two quantities, when they un- 
dergo changes of value, without altering their 
ratio. That character is oc, and is read " pro- 
portional to." Thus, if we have two quantities 
denoted by A and 15, written, 

Example. A OC B, 

the expression is read, "A proportional to B." 

Anotherkind § 2G9. There is vet another kind of relation 

of propor- . . . 

tion. which may exist between two quantities A and 
B, which it is very important to consider and 
understand. Suppose the quantities to be so 
connected with each other, that when the first 
is increased according to any law of change, the 
second shall decrease according to the same law ; 
and the reverse. 

» Olmsted's Mechanics, p. 2S. f Ibid. p. 23. 



CHAP. III.] 



GEOMETRY. 



249 




First 
Example. 



For example : the area of a rect- D 
angle is equal to the product of its 
base and altitude. Then, in the 
rectangle ABCD, we have 

Area = ABx BC. 

Take a second rectangle EFGH, having a SecoLd 
longer base EF, and a less altitude FG, but such Exam P le - 
that it shall have an equal h G 

area with the first : then we 
shall have 



Area = EFx FG. 
Now since the areas are equal, we shall have 

AB X BC = EF X FG ; Equation. 

and by resolving the terms of this equation into 
a proportion, we shall have 

AB : EF : : FG : BC. Property 

It is plain that the sides of the rectangle ABCD 
may be so changed in value as to become the 
sides of the rectangle EFGH, and that while 
they are undergoing this change, AB will in- 
crease and BC diminish. The change in the Relation? or 

the qnanU- 

values of these quantities will therefore take place ties: 
according to a fixed law : that is, one will be di- 
minished as many times as the other is increased, 



250 MATHEMATICAL SCIENCE. [BOOK II. 

since their product is constantly equal to the 
area of the rectangle EFGH. 
Expressed by Denote the side AB by x and BC by y, and 

letters. 

the area of the rectangle EFGH, which is known, 
by a ; then 

xy = a ; 

and when the product of two varying quantities 

is constantly equal to a known quantity, the two 

Reciprocal quantities are said to be Reciprocally or Inverse- 

Inverse Pro- ty proportional ; thus .r and //arc said to be in- 
p° rtl(,n - vrrsely proportional to each other. If we divide 
1 by each member of the above equation, we 
shall have 

11 

xy ~~ a • 

Reductions and by multiplying both members by x] we shall 

of the 
Equations. na\ C 

I-f. 

y « ' 

and then by dividing both numbers by x. we have 

1 

Final form. y ■ 



i 

that is, equal to the same quantity, however x or 



that is, the ratio of x to - is constantly equal to -; 



CHAP. III. J GEOMETRY. 251 



y may vary , for, a and consequently - does not 

change. Hence, 

Two quantities, which may change their values, 
are reciprocally or inversely proportional, when Proportion 

defined. 

one is proportional to unity divided by the other, 
and then their product remains constant. 

We express this reciprocal or inverse relation 
thus: 

A is said to be inversely proportional to B : the 

symbols also express that A is directly propor- How ex- 
pressed. 

tional to -^. If we have 
r> 

Aoc C' 
we say, that A is directly proportional to B, and 
inversely proportional to C. how rea ^ 

The terms Direct, Inverse or Reciprocal, ap- 
ply to the nature of the proportion, and not to 
the Ratio, which is always a mere quotient and 
the measure of proportion. The term Direct ap- Direct and 
plies to all proportions in which the terms in- j~^ 
crease or decrease together ; and the term In- applicable t* 
verse or Reciprocal to those in which one term 
increases as the other decreases. They cannot, 
therefore, properly be applied to ratio without 
changing entirely its signification and definition. 



252 



MATHEMATICAL SCIENCE. 



[book II. 



COMPARISON" OF FIGURES. 



Geometrical 
magnitudes 
compared. 



Example. 



Formula of 
Comparison. 



CbaagM of 

value : 
how affected 



Results. 



Circles com- 
pared. 



§ 270. Iii comparing geometrical magnitudes, 
by means of their quotient, it is not the quotient 
alone which we consider. The comparison im- 
plies a general relation of the magnitudes, which 
is measured by the Ratio. For example : we 
say that "Similar triangles are to each other as 
the squares of their homologous sides." What 
do we mean by that ? Just this : 

Thai the area of a triangle 

Is to the area of a similar triangle 

As the area of a square described on a side oi 

the first, 

To the area of a square described Oil all ho- 

mologous side of the second. 

Thus, we >ee that every term of Mich a pro- 
portion is in fact a surface, and that the area 

o( a triangle increases or decreases much : 
than its sides; that is, if wc double each side oi 
a triangle, the area will be four times as great : 
if we multiply each side by three, the area will 
be nine times as great ; or if we divide each 
side by two, we diminish the area four times, and 
so on. Again, 

The area of one circle 

Is to the area of another circle, 

As a square described on the diameter of the first 



CHAP. III.] 



GEOMETRY. 



253 



To a square described on the diameter of the 

second. 
Hence, if we double the diameter of a circle, How their 

areas change, 

the area of the circle whose diameter is so in- 



creased will be four times as great : if we mul- 
tiply the diameter by three, the area will be nine 
times as great ; and similarly if we divide the 
diameter. 



Principle 
general. 



Formula. 



§ 271. In comparing solids together, the same comparison 
general principles obtain. Similar solids are to 
each other as the cubes described on their ho- 
mologous or corresponding sides. That is, 

A prism 

Is to a similar prism, 
* As a cube described on a side of the first, 

Is to a cube described on an homologous side 
of the second. 

Hence, if the sides of a prism be doubled, the How lhe 

solidities 

solid contents will be increased eight-fold. Again, change. 
A sphere 

Is tO a Sphere, Sphere: 

As a cube described on the diameter of the first, 
Is to a cube described on a diameter of the 

second. 
Hence, if the diameter of a sphere be doubled, How its 

solidity 

its solid contents will be increased eight-fold ; if changes, 
the diameter be multiplied by three, the solid 



254 



M A T II E M ATICAL SCIENCE. 



[book II. 



contents will be increased twenty-seven fold : 
if the diameter be multiplied by four, the solid 
contents will be increased sixty-four fold; the 
solid contents increasing as the cubes of the 
numbers 1, 2, 3, 4, &c. 



Ratio: 



an ibetnet 

number. 

A lien hav- 
ing a lived 
value. 



§ 21'2. The relation or ratio of two magnitudes 

to each other, may he. and indeed is. expn- <1 
by an abstract number. This number has a 
fixed value so long as we d<> not introduce a 

change in the volumes of the solids; but if 
we wish t«» express their ratio under the sup- 
position that their volumes may change ac- 
cording to fixed laws (that is, so that the solids 
How rwjriag S ) K1 ]| C( )U \ j mu » similar), we then compare them 

solids are 

compared, with similar figures described on tfaeir homol* 

>us or corresponding sides; or. what is the 
same thing, take into account the corresponding 
changes which take place in the abstract num- 
bers that express their volum- 



RECAriTl'L A T I O N 



General 

outline. 



Gcoiut'tn ; 



§ 273. We have now completed a general 
outline of the science of Geometry, and what 
has been said may be recapitulated under the 
following heads. It has been shown, 

1st. That Geometry is conversant about space, 



CHAP. III.] 



GEOMETRY. 



255 



or those limited portions of space which are 
called Geometrical Magnitudes. 

2d. That the geometrical magnitudes embrace 
tnree species of figures : 

1st. Lines — straight and curved ; 
2d. Surfaces — plane and curved ; 
3d. Solids — bounded either by plane sur- 
faces or curved, or both ; and, 

4th. Angles, arising from the positions of 
lines and planes, and by which they are 
bounded. 
3d. That the science of Geometry is made up 
of those processes by means of which all the 
properties of these magnitudes are examined and 
developed, and that the results arrived at con- 
stitute the truths of Geometry. 

4th. That the truths of Geometry may be di- 
vided into three classes : 

1st. Those implied in the definitions, viz. 
that things exist corresponding to certain 
words defined ; 

2d. Intuitive or self-evident truths em- 
bodied in the axioms ; 

3d. Truths deduced (that is, inferred) from 
the definitions and axioms, called Demonstra- 
tive Truths. 
5th. That the examination of the properties of 
ihe geometrical magnitudes has reference, 



to what it 
relates. 



Lines. 

Surfaces. 

Solids. 

Angles. 



Science : 

how made 

up. 



Truths 

three classes. 

First class. 



Second. 



Third. 



Geometrica. 
magnitude*. 



256 



MATHEMATICAL SCIENCE. [BOOK II. 



comparison. 1st. To their comparison with a standard 

or unit of measure ; 

Properties. 2d. To the discovery of properties belong- 

ing to an individual figure, and yet common to 
the entire class to which such figure belongs ; 

Proportion. 3d. To the comparison, with respect to mag- 

nitude, of figures of the same species with each 
other ; viz. lines with lines, surfaces with sur- 
faces, and solids with solids. 



SUGGESTIONS FOR THOSE WHO TEACH GEOMETRY. 



{suggestions. i, ]} ( > Bure that your pupils have a clear ap- 
First. prehension of space, and of the notion that Ge- 
ometry is conversant about space only. 

2. Be sure that they understand the significa- 
Second. tion of the terms, lines, surfaces, and solids, and 

that these names indicate certain portions of 
space corresponding to them. 

3. See that they understand the distinction be- 
Third. tween a straight line and a curve ; between a 

plane surface and a curved surface ; between a 
solid bounded by planes and a solid bounded by 
curved surfaces. 

4. Be careful to have them note the charac- 
Fourth. teristics of the different species of plane figures, 

such as triangles, quadrilaterals, pentagons, hexa- 
gons, &c. ; and then the characteristic of each 






CHAP. III.] GEOMETRY. 257 

class or subspecies, so that the name shall recall, 
at once, the characteristic properties of each 
figure. 

5. Be careful, also, to have them note the 
characteristic differences of the solids. Let Fifth, 
them often name and distinguish those which 

are bounded by planes, those bounded by plane 
and curved surfaces, and those bounded by 
curved surfaces only. Regarding Solids as a 
genus, let them give the species and subspecies 
into which the solid bodies may be divided. 

6. Having thus made them familiar with the 
things which are the subjects of the reasoning, sixth, 
explain carefully the nature of the definitions ; 

then of the axioms, the grounds of our belief in 
them, and the information from which those 
self-evident truths are inferred. 

7. Then explain to them, that the definitions 

and axioms are the basis of all geometrical rea- seventh. 
soninor : that every proposition must be deduced 
from them, and that they afford the tests of all 
the truths which the reasonings establish. 

8. Let every figure, used in a demonstration, 

be accurately drawn, by the pupil himself, on a Eighth, 
blackboard. This will establish a connection 
between the eye and the hand, and give, at the 
same time, a clear perception of the figure and a 
distinct apprehension of the relations of its parts. 



258 



MATHEMATICAL SCIENCE. 



[BOOK II. 



Ninth. 



Tenth. 



9. Let the pupil, in every demonstration, first 
enunciate, in general terms, that is, without the 
aid of a diagram, or any reference to one, tho 
proposition to be proved ; and then state the 
principles previously established, which are to 
be employed in making out the proof. 

10. When in the course of a demonstration, 
any truth is inferred from its connection with 
one before known, let the truth so referred to be 
fully and accurately stated, even though the 
number of the proposition in which it is proved, 
be also required. This is deemed important. 

11. Let the pupil be made to understand that 
Eleventh, a demonstration is but a series of logical argu- 
ments arising from comparison, and that the 
result of every comparison, in respect to quan- 
tity, contains the mark either of equality or 
inequality. 

12. Let the distinction between a positive 
Tweinh. and negative demonstration be fully explained 

and clearly apprehended. 

13. In the comparison of quantities with each 
Thirteenth, other, great care should be taken to impress the 

fact that proportion exists only between quan- 
tities of the same kind, and that ratio is the 
measure of proportion. 

14. Do not fail to give much importance to 
Fourteenth, the kind of quantity under consideration. Let 



CHAP III.] GEOMETRY. 259 

the question be often put, What kind of quantity Fourteenth, 
are you considering ? Is it a line, a surface, or 
a solid ? And what kind of a line, surface, or 
solid ? 

15. In all cases of measurement, the unit of 
measure should receive special attention. If 

lines are measured, or compared by means of a Fifteenth- 
common unit, see that the pupil perceives that 
unit clearly, and apprehends distinctly its rela- 
tions to the lines which it measures. In sur- 
faces, take much pains to mark out on the 
blackboard the particular square which forms 
the unit of measure, and write unit, or unit of 
measure, over it. So in the measurement of 
solidity, let the unit or measuring cube be ex- 
hibited, and the conception of its size clearly 
formed in the mind ; and then impress the im- 
portant fact, that, all measurement consists in 
merely comparing a unit of measure with the 
quantity measured; and that the number which 
expresses the ratio is the numerical expression 
for that measure. 

16. Be careful to explain the difference of the 
terms Equal and Equivalent, and never permit sixteenth, 
the pupil to use them as synonymous. An ac- 
curate use of words leads to nice discriminations 

of thought. 



CHAP. IV.] 



AN AL Y SI3. 



261 



CHAPTER IV. 

ANALYSIS — ALGEBRA — ANALYTICAL GEOMETRY DIFFERENTIAL AND INTEGRAL 

CALCULUS. 

ANALYSIS. 

§ 274. Analysis is a general term, embra- Analysis 
eing that entire portion of mathematical science 
in which the quantities considered are repre- 
sented by letters of the alphabet, and the opera- 
tions to be performed on them are indicated by 
sip-vis. 



Numbers 

must be ol 

things ; 



§ 275. We have seen that all numbers must 
be numbers of something ; # for, there is no such 
thing as a number without a basis : that is, every 
number must be based on the abstract unit one, 
or on some unit denominated. But although 
numbers must be numbers of something, yet they but may be 

. . °f many land 

may be numbers of any thing, for the unit may of things. 
be whatever we choose to make it. 



* Section 112. 



262 MATHEMATICAL SCIENCE. [fiOOK II. 

aii quantity § 276. All quantity consists of parts, which 

consists of . . . ... 

p arts# can be numbered exactly or approximatively, 

and, in this respect, possesses all the properties 

of number. Propositions, therefore, concerning 

numbers, have the remarkable peculiarity, that 

Propositions they are propositions concerning all quantities 

in regard to 

number whatever. That half of six is three,, is equally 

jpply also to , , . 

quantity. true > whatever the word six may represent, 
whether six abstract units, six men, or six tri- 
angles. Analysis extends the generalization still 
further. A number represents, or stands for, that 
particular number of things of the same kind, 

Algebraic without reference to the nature of the thing; 

symbols 

more gener- but an analytical symbol does more, for it may 
stand for all niunbcrs, or for all quantities which 
numbers represent, or even for quantities which 
cannot be exactly expressed numerically. 

Anything As soon as we conceive of a thing we may 

conceived . ........ , , 

maybedi- conceive it divided into equal parts, and may 

vuled ' represent either or all of those parts by a or x, 

or may, if we please, denote the thing itself by a 

or x, without any reference to its being divided 

into parts. 



Each figure §277. In Geometry, each geometrical figure 

stands for a , r , , , , , 

stands lor a class ; and when we have demon- 



class. 



strated a property of a figure, that property is 
considered as proved for every figure of the class. 



CHAP. IV.] ANALYSIS. 263 

For example : when we prove that the square Example, 
described on the hypothenuse of a right-angled 
triangle is equivalent to the sum of the squares 
described on the other two sides, we demonstrate 
the fact for all right-angled triangles. But in 
analysis, all numbers, all lines, all surfaces, all in analysis 

the symbols 

solids, may be denoted by a single symbol, a or x. stand for 

Hence, all truths inferred by means of these classes. 

symbols are true of all things whatever, and not 

like those of number and geometry, true only 

of particular classes of things. It is, therefore, 

not surprising, that the symbols of analysis do 

not excite in our minds the ideas of particular Hence, the 

. . . truths infer- 

thmgs. lhe mere written characters, a, o, c, a, red 
x, y, z, serve as the representatives of things in 
general, whether abstract or concrete, whether 
known or unknown, whether finite or infinite. 



§ 278. In the uses which we make of these symbols 

come to be 

symbols, and the processes of reasoning carried regarded a/ 
on by means of them, the mind insensibly comes 
to regard them as things, and not as mere signs ; 
and we constantly predicate of them the prop- 
erties of things in general, without pausing to 
inquire what kind of thing is implied. Thus, Example, 
we define an equation to be a proposition in Theequa- 
which equality is predicated of one thing as tl0n * 
compared with another. For example : 



are gen 
eral. 



264 MATHEMATICAL SCIENCE. [bOOKII 



a + c = x, 
vvhataxioms * s au equation, because z is declared to be 

necessary to e q Ua J to f} ie sum f a an( l c J^ t ] ie solution of 
its solution. A 

equations, we employ the axioms, "If equals be 

added to equals, the sums will be equal ;" and, 

" If equals be taken from equals, the remainders 

They express will be equal.'' Now, these axioms do not ex- 
qualities Of ■••*•! • 

things, press qualities of language, but properties of 
Hence, in- quantity. Hence, all inferences in mathemat- 

ferences re- 

late to things. ICal science, deduced through the instrumentality 

of symbols, whether Arithmetical, Geometrical, 

or Analytical, must be regarded as concerning 

quantity, and not symbols. 

Quantity A.s analytical symbols are the representatives 
neednota- j quantity in general, there is no necessity of 

whys be pnt" ■ 

enttothe (seeping the idea of quantity continually alive in 
the mind: and the processes of thought may, 
without danger, be allowed to rest on the sym- 
bols themselves, and therefore, become to that 

extent, merely mechanical. But, when we look 
The reason- back and see on what the reasoning is based, and 
based on the now tne processes have been conducted, we shall 
supposition fj n( j t ] lat evcrv ste p was taken on the supposition 

of quantity. J i n 

that we were actually dealing with things, and 
not symbols; and that, without this understand- 
ing of the language, the whole system is without 
signification, and fails. 



CHAP. IV.] ALGEBRA. 205 



§ 279. There are three principal branches of Three 

A , . branches 

Analysis : 

1st. Algebra. Algebra, 

2d. Analytical Geometry. Analytical 

Geometry, 

3d. Differential and Integral Calculus. mm m. 



ALGEBRA. 

§ 280. Algebra is, in fact, a species of uni- Algebra: 
versal Arithmetic, in which letters and signs are universal 
employed to abridge and generalize all processes * me lCy 
involving numbers. It is divided into two parts, two parts 
corresponding to the science and art of Arith- 
metic : 

1st. That which has for its object the investi- First part, 
gation of the properties of numbers, embracing 
all the processes of reasoning by which new 
properties are inferred from known ones ; and, 

2d. The solution of all problems or questions second part 
involving the determination of certain numbers 
which are unknown, from their connection with 
certain others which are known or given. 



ANALYTICAL GEOMETRY 



§ 281. Analytical Geometry examines the Analytical 
properties, measures, and relations of the geo- 
metrical magnitudes by neans of the analytical it* nature. 



266 MATHEMATICAL SCIENCE. [BOOK II 



symbols. This branch of mathematical science 
Descartes, originated with the illustrious Descartes, a cele- 

the original 

founder of brated French mathematician of the 17th cen- 

this science. TT , , . . . . . 

tury. He observed that the positions 01 points, 
observed, the direction of lines, and the forms of surfai 

could be expressed by means of the algebraic 
aii position symbols ; and consequently, that every chai 

expressed by 

symbols, either in the position or extent of a geometrical 
magnitude, produced a corresponding change in 
certain symbols, by which such magnitude could 
be represented. A- soon as ii was found that, 

to every variety of position in points, direction 
in lines, or form of curves or surfaces, th. 
responded certain analytical expressions (called 
their Equations), ii followed, that if the processes 
were known by which these equations could be 
Theequation examined, the relation of their parts determined, 

develops the ' . 

properties and the laws according to which those parts 
° nitude. ag " vary, relative to one another, ascertained, then 
the corresponding changes in the geometrical 
magnitudes, thus represented, could be imme- 
diately inferred. 

Hence, it follows that every geometrical ques- 
Powerover tion can be solved, if we can resolve the corre- 

the magni- . 

tude extend- spondmg algebraic equation ; and the power over 

edbythe t ^ e g eome t r i C al magnitudes was extended just in 

proportion as the properties of quantity were 

brought to light by means of the Calculus. The 



CHAP. IV.] ANALYSIS. 2()7 

applications of this Calculus were soon made to To what sub- 

. , . ject applied. 

the subjects of mechanics, astronomy, and in- 
deed, in a greater or less degree, to all branches 
of natural philosophy; so that, at the present its present 
time, all the varieties of physical phenomena, 
with which the higher branches of the science 
are conversant, are found to answer to varieties 
determinable by the algebraic analysis. 



§ 282. Two classes of quantities, and conse- Quantities 

which enter 

quently two sets of symbols, quite distinct from into the cai- 
each other, enter into this Calculus ; the one 
called Constants, which preserve a fixed or given constants. 
value throughout the same discussion or investi- 
gation ; and the other called Variables, which variables, 
undergo certain changes of value, the laws of 
which are indicated by the algebraic expressions 
or equations into which they enter. Hence, 

Analytical Geometry may be defined as that Analytical 
branch of mathematical science, which exam- defined." 
ines, discusses, and develops the properties of 
geometrical magnitudes by noting the changes 
that take place in the algebraic symbols which 
represent them, the laws of change being deter- 
mined by an algebraic equation or formula. 



208 MATHEMATICAL SCIENCE. [bOOKIL 



DIFFERENTIAL AND INTEGRAL CALCULUS. 

Quantities § 2 83. In this branch of mathematical science, 

considered. 

as in Analytical Geometry, two kinds of quan- 
vambies, t j ty are consider^ yiz. Variables and Constants ; 

Constants. 

and consequently, two distinct sets of symbols 
The Science, arc employed. The science consists of a series 
of processes which note the changes that I 
place in the value of the Variables. Those 
changes of value take place according to fixed 
laws established by algebraic formulas, and are 

Marks. indica te< I by certain marks drawn from the va- 
Difiwentiai riable symbols, called Differential I nts. 

By these marks we art enabled to trace out with 

the accuracy of exact science the most hidden 

properties of quantity, as well as the most gen- 
eral and minute laws which regulate its chan_ 
of value. 



Coefficient*. 



Analytical § 284. It will be observed, that Analytical 

and Geometry and the Differential and Integral Cal- 

Caicuius: cu i us treat of quantity regarded under the same 

general aspect, viz. as subject to changes or va- 

Howtiey nations in magnitude according to laws indica- 

regard quan- t 

tity : ted by algebraical formulas; and the quantities, 

whether variable or constant, are, in both cases, 

by what represented by the same algebraic symbols, viz. 

represented. ^ cons tants by the first, and the variables by 



CHAP. IV.] ALGEBRA. 269 

the final letters of the alphabet. There is, how- Difference; 
ever, this important difference : in Analytical 
Geometry all the results are inferred from the in* hat it 

consists. 

lelations which exist between the quantities 
themselves, while in the Differential and Integral 
Calculus they are deduced by considering what 
may be indicated by marks drawn from variable 
quantities, under certain suppositions, and by 
marks of such marks. 

§ 285. Algebra, Analytical Geometry, the Dif- Analytical 

Science. 

ferential and Integral Calculus, extended into the 
Theory of Variations, make up the subject of 
analytical science, of which Algebra is the ele- 
mentary branch. As the limits of this work do its parts, 
not permit us to discuss the subject in full, we 
shall confine ourselves to Algebra, pointing out, 
occasionally, a few of the more obvious connec- How far 
tions between it and the two other branches. 



ALGEBRA . 

. § 286. On an analysis of the subject of Alge- Algebra, 
bra, we think it will appear that the subject itself 
presents no serious difficulties, and that most of Difficulties, 
the embarrassment which is experienced by the 
pupil in gaining a knowledge of its principles, as H ow over- 
well as in their applications, anses from not at 



270 MATHEMATICAL SCIENCE. [BOOK II. 

Language, tending sufficiently to the language or signs of 
the thoughts which are combined in the reason- 
ings. At the hazard, therefore, of being a little 
diffuse, I shall begin with the very elements of 
the algebraic language, and explain, with much 
minuteness, the exact signification of the char- 

Chanden acters that stand for the quantities which are the 

which repre- , . , . . . . , 

eentquontity. subjects ot the analysis ; and also ot those signs 
signs. which indicate the several operations to be per- 
formed on the quantities. 

Qualities. § 287. The quantities which are the subjects 

now divided, of the algebraic analysis may be divided into 

two classes: those which are known or given, 

and those which are unknown or sought. The 

How repre- known are uniformly represented by the first 
letters of the alphabet, a, />, c, ch &c. ; and the 
unknown by the final letters, x s y, X, D, 10, &C 



May be in- 
creased or 



§ 288. Quantity is susceptible of being in- 
creased or diminished ;* and there are five opcr- 

diminished. 

Five opera- ations which can be performed upon a quantity 
tiom ' that will give results differing from the quantity 

itself, viz. : 
p . gt 1st. To add it to itself or to some other quan- 

tity; 



* Section 75. 



CHAP. IV J ALGEBRA, 271 

2d. To subtract some other quantity from it ; second. 

3d. To multiply it by a number ; Third. 

4th. To divide it ; Fourth. 

5th. To extract a root of it. Fifth - 

The cases in which the multiplier or divisor 

is 1, are of course excepted ; as also the case Exception, 
where a root is to be extracted of 1. 

§ 289. The five signs which denote these oper- signs. 
ations are too well known to be repeated here. 

These, with the signs of equality and inequalitv, Elements 

of the 

the letters of the alphabet and the figures which Algebra.: 

are employed, make up the elements of the alge- anguage ' 

braic language. The words and phrases of the its words 

algebraic, like those of every other language, are 

to be taken in connection with each other, and 

are not to be interpreted as separate and isolated How inter- 
preted. 
symbols. 

§ 290. The symbols of quantity are designed symbols ci 

to represent quantity in general, whether abstract quantl> * 
or concrete, whether known or unknown; and 

the signs which indicate the operations to be General, 
performed on the quantities are to be interpreted 
in a sense equally general. When the sign plus 

is written, it indicates that the quantity before Examples, 

which it is placed is to be added to some other signs plus 

quantity ; and the sign minus implies the ey 1 ^ ^ mmU3, 



272 MATHEMATICAL SCIENCE. [euOK II 

ence of a minuend, from which the subtrahend 

is to be taken. One thing should be observed in 

signs have regard to the signs which indicate the operation* 

no effect on 

the nature of that are to be performed on quantities, viz. the\ 

a quantity. 

do not at all affect or change the nature of th 
quantity before or after which they are writttX 
but merely indicate what is to be done with tht 
Example*: quantity. Iii Algebra, for example., the minus 
in Algebra. ^^ merely indicates that the quantity before 
which it is written is to be subtracted from 

in Analytical some other quantity ; and in Analytical Geom- 

Geometry. ........ 

etry, that the line before which it lulls ifl esti- 
mated in a contrary direction from that in which 

it would have been reckoned, had it had the sign 
plus; but in neither case i> the nature of the 

quantity itself different from what it would have 
been had it had the sign plus. 
interprets The interpretation of the language of Algebra 

language? > s tne f ,l ' st thing U) which the attention of a pupil 
should be directed ; and he should be drilled on 
the meaning and import of the symbols, until 
their significations and uses are as familiar as 
its necessity, the sounds and combinations of the letters of the 
alphabet. 

Elements § 291. Beginning with the elements of the 
expamc . j an g ua g e ^ \ e ± an y nun iber or quantity be desig- 
nated by the letter a, and let it be required to 



CHAP. IV. J ALGEBRA. 273 



add this letter to itself, and find the result or sum. 
The addition will be expressed by 

a + a — the sum. 

But how is the sum to be expressed ? By simply signification 
regarding a as one a, or la, and then observing 
that one a and one a make two as or 2 a: hence, 

a + a = 2 a ; 

and thus we place a figure before a letter to in- 
dicate how many times it is taken. Such figure 
is called a Coefficient. Coefficient 

§ 292. The product of several numbers is in- Product: 
dicated by the sign of multiplication, or by sim- 
ply writing the letters which represent the num- 
bers by the side of each other. Thus, 

a x b x c x d xf, or abcdf, how in<iic» 

ted 

indicates the continued product of a, b, c, d, and 
f, and each letter is called a factor of the prod- 
uct : hence, a factor of a product is one of the r***. 
multipliers which produce it. Any figure, as 5, 
written before a product, as 

5 abcdf, 

is the coefficient of the product, and shows that coefficient oi 

the product is taken 5 times. ap 

18 



274 M A T H E M ATICAL S C 1 E N C E . | BOOK II. 



Equal fac- § 293. If the numbers represented by a, b. c, 

a, and f were equal to each other, they would 

what the each be represented by a single letter a. and the 

product ' 

become, product wcu.d then become 



iiow 

expressed 



a x a x a x a x a = a ° ] 

that is,, we indicate the product of several equal 
factors by simply writing the letter once and 
placing a figure above and a little at the right 

of it, to indicate how many times it is taken as 

Exponent: ; , factor. The figure BO written is called an 

where writr exponent Hence, an exponent is a simple form 
of expression, to point out how many equal fac 
tors are employed. 

Division-. §294. The division of one quantity by an- 

how other is indicated by simply writing the divisor 

expressed. 

below the dividend, after the manner 01 a frac- 
tion : by placing it on the right of the dividend 
with a horizontal line and two dots between them ; 

or by placing it on the right with a vertical line 
between them : thus cither form of expression : 

Three lorms. J» * "?? «| Pf 1> \ a 

indicates the division of b by a. 

Koots: § 295. The extraction of a root is indicated 

k>w indie* by the sign </. This sign, when used bv itself 

indicates the lowest root, viz. the square root. 



CHAP. IV.] ALGEBRA. 275 

If any other root is to be extracted, as the third, 

fourth, fifth, &c, the figure marking the degree index; 

of the root is written above and at the left of where writ- 
ten, 
the sign ; as, 

*\T~ cube root, \/~ fourth root, &c. 

The figure so written, is called the Index of the 
root. 

We have thus given the very simple and gen- Language 

. . i l • i • i • for the five 

eral language by which we indicate every one operation* 
of the five operations that may be performed on 
an algebraic quantity, and every process in Al- 
gebra involves one or other of these operations. 



MINUS SIGN. 

§ 296. The algebraic symbols are divided into Algebraic 

language ; 

two classes entirely distinct from each other, 

viz. the letters that are used to designate the how divided. 

quantities which are the subjects of the science, 

and the signs which are employed to indicate 

certain operations to be performed on those 

quantities. We have seen that all the algebraic Algebraic 

processes are comprised under addition, subtrac- 

A x their num- 

tion, multiplication, division, and the extraction ker. 



of roots; and it is plain, that the nature of a Donot 
quantity is not at all changed by prefixing to it nau^of th 
the sign which indicates either of these opera- ( i uaDtltie8 - 



276 MATHEMATICAL SCI E \CE. [BOOK II. 

tions. The quantity denoted by the letter a, for 
example, is the same, in even/ respect, whatever 
sign maybe prefixed to it: that is, whether it 
be added to another quantity, subtracted from 
it, whether multiplied or divided by any number, 
or whether we extract the square or cube or any 
Algebraic other root of it. The algebraic signs, therefore* 

signs: 

how regard- must be regarded merely as indicating 
turns to be pe i on quantity, Hid 

affecting the iKthwr <>f the quantities to which 

they may be prefixed. We say, indeed, that 
Phis and quant ities are plus and minus, but this is an ah- 

Mfaras. breviated language to express that they are tn 
be added or subtracted. 

Principles of §297. In Algebra, as in Arithmetic and Ge- 

• ■ •' oinetry. all the principles of the science are de- 
duced from the definitions and axioms ; and the 
rules for performing the operations are but di- 
rections framed in conformity to such principles 
Having, for example, fixed by definition, the power 
of the minus sign, viz. that any quantity before 
which it is written, shall be regarded as to be 
subtracted from another quantity, we wish to 
wishtodis- di SCO ver the process of performing that subtrac- 
tion, so as to deduce therefrom a general purine 
ciple, from which we can frame a rule applicable 
to all similar cases. 



From what 

deduced. 



Example. 



What \vc 



cover. 



CHAT. IV.] 



ALGEBRA, 



277 



SUBTRACTION, 



b 

a- 



Process. 



§ 298. Let it be required, for example, to subtraction. 
subtract from b the difference be- 
tween a and c. Now, having writ- 
ten the letters, with their proper 
signs, the language of Algebra expresses that it 
is the difference only between a and c, which is 
to be taken from b ; and if this difference were Difference, 
known, we could make the subtraction at once. 
But the nature and generality of the algebraic 
symbols, enable us to indicate operations, merely, operation! 

. . iii« .1 indicated. 

and we cannot in general make reductions until 
we come to the final result. In w T hat general 
way, therefore, can we indicate the true differ- 
ence ? 

If we indicate the subtraction of 
a from b, we have b — a; but then 
we have taken away too much from 
b by the number of units in c, for it was not a, 
but the difference between a and c that was to 
be subtracted from b. Having taken away too 
much, the remainder is too small by c : hence, 
if c be added, the true remainder w T ill be express- 
ed by b — a + c. 

Now, by analyzing this result, we see that the Analysis a 

r the result. 

sign ot every term of the subtrahend has been 
changed ; and what has been shown with re- 



ft— ^ 

b — a + c 



Final 
formula 



278 



M A T H EMATICAL SCIENCE. 



[book II. 



Generaiiza- spect to these quantities is equally true of all 
others standing in the same relation : hence, we 
have the following general rule for the subtrac- 
tion of algebraic quantities : 

Change the sign of evert/ term of the subtra- 
Ru,e - hend, or conceive it to be changed, and then unite 
the quantities as in addition. 



Multiplica- 
tion. 



Signification 
of the 
language. 



ac—bc 



MULTIPLICATION. 

§299. Let us now consider the case of mul- 
tiplication, and let it be required to multiply 
a — b by r. The algebraic language expn 
that the difference between a and b 

is to be taken as many times as 
there are units in c. If we knew 
this difference, we could at oner 
perform the multiplication. But by what gen- 
I eral process is it to be performed without finding 

that difference ? If we take a, c times, the prod-; 
net will be ac ; but as it was only the difference 
between a and b, that was to be multiplied by <\ 
this product ac will be too great by b taken c 
times ; that is, the true product will be expressed 
by ac — bc: hence, we see, that, 

If a quantity having a plus sign be multi- 
plied by another quantity having also a plus 
sign, the sign of the product will be plus : and 



Tte nature. 



\ tnciple for 

the sigr9. 



CHAP. IV.] \LGEBRA. 279 

if a quantity having a minus sign be multi- 
plied by a quantity having a plus sign, the sign 
of the product will be minus. 

§ 300. Let us now take the most general General case 
case, viz. that in which it is required to multi- 
ply a — b by c — d. 

Let us again observe that the algebraic lan- 
guage denotes that a — b is 
to be taken as many times 
as there are units in c— d\ 
and if these two differences 
were known, their product 



a — b 

C — d I* 8 form. 



ac — be 

— ad-\-bd 

ac — bc — ad + bd 



would at once form the product required. 

First : let us take a — b as many times as there nrst step, 
are units in c ; this product, from what has al- 
ready been shown, is equal to ac — be. But 
since the multiplier is not c, but c — d, it follows 
that this product is too large by a — b taken d 
times ; that is, by ad — bd: hence, the first prod- second step 
uct diminished by this last, will give the true 
product. But, by the rule for subtraction, this 
difference is found by changing the signs of the HowtakeQ - 
subtrahend, and then uniting all the terms as in 
addition : hence, the true product is expressed 
by ac — be — ad + hd. 

By analyzing this result, and employing an Analysis of 
abbreviated language, we have the following gen- 



280 MATHEMATICAL SCIENCE. [BOOK II. 

eral principle to which the signs conform in mul- 
tiplication, viz. : 
General Plus multiplied by plus gives plus in the prod- 

nncipe. uc ^ . pi us multiplied by minus gives minus ; mi* 
nus multiplied by plus gives minus; and minus 
multiplied by minus gints plus in the product. 

Remark. § 301. The remark is often made by pupils 
that the above reasoning appears very satisfac- 
tory so long as the quantities are presented un- 
der the above farm : but why will —h multiplied 

Particular ] )y __,/ gj ve p[ us J )( J p H ow can t | R > product <>t 
case. 

two negative quantities standing alone be plus ? 
Nfawriga: in tll(k lirst l ,la(v - lllr i^hms sign being pre- 
fixed to b and ti shows that in an algebra! 

they do not stand by themselves, but arc con- 
iteinierpre- nected with other quantities; and if they are 

lalion. . . 

not so connected, the minus sign makes no dll- 

ference ; lor.it in no ease affects the quantity, 

but merely points out a connection with other 

quantities. Besides, the product determined 

above, being independent of any particular value 

attributed to the letters a, b, c, and d, must be 

Form of the of such a form as to be true for all values; and 

miTbelrue hence for the case in which a and c are both 

for quantities a j to zero Making this supposition, tlie 

of any value. 1 ° 

product reduces to the form of + bd. The roles 
for the signs in division are readily deduced from 



CHAP. IV.] 



ALGEBRA. 



281 



the definition of division, and the principles al- signs in 
ready laid down. 



ZERO AND INFINITY. 



§ 302. The terms zero and infinity have given zero and 

Infinity. 

rise to much discussion, and been regarded as 
presenting difficulties not easily removed. It may 
not be easy to frame a form of language that shall 
convey to a mind, but little versed in mathe- 
matical science, the precise ideas which these 
terms are designed to express ; but we are un- 
willing to suppose that the ideas themselves arc 
beyond the grasp of an ordinary intellect. The 
terms are used to designate the two limits of 
Space and Number. 



Ideas not 
abstruse. 



§ 303. Assuming any two points in space, and 
joining them by a straight line, the distance be- 
tween the points will be truly indicated by the 
length of this line, and this length may be ex- 
pressed numerically by the number of times 
which the line contains a know r n unit. If now, 
the points are made to approach each other, the illustration, 

showing tho 

length of the line will diminish as the points meaning^ 
come nearer and nearer together, until at length, Zero< 
when the two points become one, the length of 
the line will disappear, having attained its limit, 



28-2 



MATHEMATICAL SCIENCE. 



[book II 



which is called zero. If, on the contrary, the 

points recede from each other, the length of the 

illustration, ]j Iie joining them will continually increase ; hut 

showing the ' 

meaning of so long as the length oi the line can be expres 

in terms of a known unit of measure, it is not 
infinite. But, if we suppose the points removed, 
so that any known unit of measure would occupy 
no appreciable portion of the line, then the length 
of the line is said to he Infinite. 



the term 

Infinity. 



§301. Assuming one as the unit of number, 
and admitting the self-evident truth that it may 
he increased or diminished, we shall have no 

Th,,l,,,ms difficulty in understanding the import of the 

Zero and In- . * 

Unity applied tlTms Kero anc | infinity, as applied to numher. 
to cumbers. 

For, if we suppose the unit one to be continually 
diminished, by division or otherwise, the firac- 

niustr.ition. tional units thus arising will he le>s and less, 
and in proportion as we continue the divisions, 
they will continue to diminish. Now. the limit 
or boundary to which these very -mall fractions 
zero: approach, is called Zero., or nothing. So long 
as the fractional number forms an appreciable 
part of one, it is not zero, but a finite fraction : 
and the term zero is only applicable to that 
which forms no appreciable part of the standard. 

illustration. If, on the other hand, we suppose a numbei 
to be continually increased, the relation of this 



CHAP. IV. "I ALGEBRA. 283 

number to the unit will be constantly changing. 
So long as the number can be expressed in 
terms of the unit one. it is finite, and not infi- infinity; 
nite ; but when the unit one forms no appre- 
ciable part of the number, the term infinite is 
used to express that state of value, or rather, 
that limit of value. 

§ 305. The terms zero and infinity are there- The terms, 
fore employed to designate the limits to which emp i ove< i. 
decreasing and increasing quantities may be 
made to approach nearer than any assignable 
quantity ; but these limits cannot be compared, Are i imit& 
in respect to magnitude, with any known stand- 
ard, so as to give a finite ratio. 



Whv limits! 



§ 306. It may, perhaps, appear somewhat par- 
adoxical, that zero and infinity should be defined 
as u the limits of number and space'' when they 
are in themselves not measurable. But a limit 
is that " which sets bounds to, or circumscribes ;'* Definition of 
and as all finite space and finite number (and aUm,t * 
such only are implied by the terms Space and orspaceand 
Number), are contained between zero and in- Nambcr 
finity, we employ these terms to designate the 
limits of Number and Space. 



284 MATHEMATICAL SCIENCE. [BOOK II. 



OF THE EQUATION. 

Deductive § 307. We have seen that all deductive rea- 

reasonuig. 

soning involves certain processes of comparison, 

and that the syllogism is the formula to which 

those processes may be reduced.* It has also 

comparison b een stated that if two quantities be compared 

of quantities. k ■ 

together, there will necessarily result the condi- 
Condition. tion of equality or inequality. The equation is 
an analytical formula for expressing equality. 

sui.jcct of $ 308 T he sul)jrct f equation* j s divided 

equations: 

howdhided. into two parts. The first, consists in find 

First part: the equation ; that is. in the process of expi 

ing the relations exist iiiLT between the quantities 

considered, by means of the algebraic symbols 

statement, and formula. This is called the Statement of 

second part: the proposition. 'Flie second is purely deduc- 
tive, and consists, in Alircbra, in what is called 
Solution. the solution of the equation, or finding the value 
of the unknown quantity ; and in the other 
branches of analysis, it consists in the dtsone- 

niscussion of sion of the equation ; that is, in the drawing out 
from the equation every thing which it is ca- 
pable of expressing. 

* Section 98. 



CHAP. IV.] ALGEBRA. 285 

§ 309. Making the statement, or finding the summ*: 
equation, is merely analyzing the problem, and what it* 
expressing its elements and their relations in 
the language of analysis. It is, in truth, col- 
lating the facts, noting their bearing and con- 
nection, and inferring some general law or prin- 
ciple which leads to the formation of an equation. 

The condition of equality between two quan- Equality of 

tities is expressed "by the sign of equality, which ^^mz 

is placed between them. The quantity on the How ex- 
pressed, 
left of the sign of equality is called the first mem- lstmember 

ber, and that on the right, the second member ?llinnitfirr 
of the equation. The first member corresponds 
to the subject of a proposition ; the sign of equal- subject. 
ity is copula and part of the predicate, signify- Predicate, 
ing, is equal to. Hence, an equation is merely 
a proposition expressed algebraically, in which Pr position, 
equality is predicated of one quantity as com- 
pared with another. It is the great formula o( 
analysis. 

§ 310. We have seen that every quantity is Abstract, 
either abstract or concrete :* hence, an equa- concrete, 
tion, which is a general formula for expressing 
equality, must be either abstract or concrete. 

An abstract equation expresses merely the 

* Section 75. 



286 



MATHEMATICAL SCIENCE. [BOOK II 



relation of equality between two abstract quan 
tities : thus, 

a + b = x, 



Abstract 
equation. 



Concrete 
equation. 



is an abstract equation, if no unit of value be 
assigned to either member ; for, until that be 
done the abstract unit one is understood, and tiie 
formula merely expresses that the sum of a and /; 
is equal to x, and is true, equally, of all quantities. 
But if we assign a concrete unit of value, that 
is, say that a and b shall each denote so many 
pounds weight, or so many feet or yards of 
length, x will be of the same denomination, and 
the equation will become concrete or denominate. 



Five open- § 311. We have seen that there are five oper- 
perfbnned. ations which may be performed on an algebraic 
quantity.* We assume, as an axiom, that if 
the same operation, under either of these pro- 
cesses, be performed on both members of an 
equation, the equality of the members will not be 
changed. Hence, we have the five following 



Axioms. AXIOMS. 

First. 1. If equal quantities be added to both mem- 

bers of an equation, the equality of the members 
will not be destroyed. 

* Section 288. 



CHAP. IV.] 



ALGEBRA. 



2S7 



2. If equal quantities be subtracted from both c ecomJ 
members of an equation, the equality will not be 
destroyed. 

3. If both members of an equation be multi- 
plied by the same number, the equality will not 
be destroyed. 

4. If both members of an equation be divided 
by the same number, the equality will not be 
destroyed. 

5. If the same root of both members of an 
equation be extracted, the equality of the mem- 
bers will not be destroyed. 

Every operation performed on an equation 
wiD fall under one or other of these axioms, and 
they afford the means of solving all equations 
which admit of solution. 



Third. 



Fourth. 



Fifth. 



Use of 

axioms. 



§ 312. The term Equality, in Geometrv, ex- Equality: 

Its meaning 

presses that relation between two magnitudes in Geometry 

which will cause them to coincide, throughout 

their whole extent, when applied to each other. 

The same term, in Algebra, merely implies that its meaning 

in Algebra. 

the quantity, of which equality is predicated, 
and that to which it is affirmed to be equal, 
contain the same unit of measure an equal num- 
ber of times : hence, the algebraic signification 
of the term equality corresponds to the signi- corresponds 

to equiva- 

fication of the geometrical term equivalency. lency. 



288 MATHEMATICAL SCIENCE. [bOOKII. 

§ 313. We have thus pointed out some of the 

marked characteristics of analysis. In Algebra, 

ciassesof the elementary branch, the quantities, about 

quantities ii\ . . , . ..... 

Algebra, which the science is conversant, are divided, 
as has been already remarked, into known and 
unknown, and the connections between them. 
expressed by the equation, afford the mean- 
tracing out farther relations, and of finding the 
values of the unknown quantities in terms of the 
known. 

In the other branches of analysis, the quanti- 
iiow divided tj cs considered arc divided into two general 

in the other 

branch^ of closoeo, Constant and Variable: the former pre- 

Analv^K 

serving fixed vwues throughout the same pro- 
ii of investigation, while the latter unde 

changes of value according to fixed laws: and 

from such changes we deduce, by means of the 

equation, common principles, and general prop- 
erties applicable to all quantit: 

Correspond- § 314. The correspondence between the pr<>- 

ence in 

methods of cesses of reasoning, as exhibited in the subject of 

■Mounted general logic, and those which are employed in 

,or * mathematical science, is readily accounted for, 

when we reflect., that the reasoning process is 

essentially the same in all cases: and that any 
change in the language employed, or in the sub- 
ject to which the reasoning is applied, does not 



CHAP. IV.] ALGEBRA. 289 

at all change the nature of the process, or mate- 
rially vary its form. 

§ 315. We shall not pursue the subject of 
analysis any further; for, it would be foreign 
to the purposes of the present work to attempt objects of 

j the present 

more than to point out the general teatures and work: 
characteristics of the different branches of math- 
ematical science, to present the subjects about 
which the science is conversant, to explain the 
peculiarities of the language, the nature of the 
reasoning processes employed, and of the con- 
necting links of that golden chain which binds extende d. 
together all the parts, forming an harmonious 
whole. 



SUGGESTIONS FOR THOSE WHO TEACH ALGEBRA. 

1. Be careful to explain that the letters em- Letters are 
ployed, are the mere symbols of quantity. That Emboli 
of, and in themselves, they have no meaning or 
signification whatever, but are used merely as 

.he signs or representatives of such quantities 
as they may be employed to denote. 

2. Be careful to explain that the signs which signs indi- 

are used are employed merely for the purpose tiona. 

of indicating the five operations which may be 

performed on quantitv ; and that thev indicate 

19 



290 MATHEMATICAL SCIENCE. [BOOK II. 

operations merely, without at all affecting the 

nature of the quantities before which they are 

placed. 

Letters and 3. Explain that the letters and signs are the 

eieme^L of e l ements °f ^ ie algebraic language, and that the 

language, language itself arises from the combination 01 

these elements. 
Algebraic 4. Explain that the finding of an algebraic 
formula is but the translation of certain id 
first expressed in our common language, into 
the language of Algebra ; and that the interpre- 
ita interpret- tation of an algebraic formula is merely trans 

ation. 

lating its various significations into common 
languag 

Language, 5. Let the language of Algebra be carefully 

studied, so that its construction and significa- 
tions may be clearly apprehended. 
coefficient, 6. Let the difference between a coefficient 

Exponent and an exponent be carefully noted, and the 
office of each often explained ; and illustrate ire 
quently the signification of the language by at- 
tributing numerical values to letters in various 
algebraic expressions, 
similar 7, Point out often the characteristics of sim- 

quantities. 

ilar and dissimilar quantities, and explain which 
may be incorporated and which cannot. 
Minus sign. 8. Explain the power of the minus sign, as 
shown in the four ground rules, hut very par- 



CHAP. IV.] ALGEBRA. 291 

ticularly as it is illustrated in subtraction and 
multiplication. 

9. Point out and illustrate the correspondence 
between the four ground rules of Arithmetic Arithmetic 

and Algebra 

and Algebra; and impress the fact, that their compared, 
differences, wherever they appear, arise merely 
from differences in notation and language : the 
principles w r hich govern the operations being 
the same in both. 

10. Explain with great minuteness and par- Equation. 
ticularity all the characteristic properties of the ** proper 

ties. 

equation ; the manner of forming it ; the differ- 
ent kinds of quantity which enter into its com- 
position ; its examination or discussion ; and 
the different methods of elimination. 

11. In the equation of the second degree, be Equation oi 
careful to dwell on the four forms which em- degree. 
brace all the cases, and illustrate by many ex- 
amples that every equation of the second de- 
gree may be reduced to one or other of them, its forms. 
Explain very particularly the meaning of the 

term root ; and then show, why every equation lts roots - 
of the first degree has one, and every equation 
of the second degree two. Dwell on the prop- 
erties of these roots in the equation of the sec- 
ond degree. Show why their sum, in all the Their sum, 
forms, is equal to the coefficient of the second 
term, taken with a contrary sign ; and why their 



292 MATHEMATICAL SCIE X C E . [BOOK II. 



Their prod- product is equal to the absolute term with a 
contrary sign. Explain when and why the roots 



General \o. In fine, remember that e operation 

Principles : 

and rule is based on a principle of science, and 
that an intelligible reason may be given for it. 
Find that and impress it on the mind 

3houUt be of you? pupil in plain and simple lai and 

by familiar and appropriate illustration*. You 
will thus impr est right habits of investigation 
and study, and he will <zrow in knowledge. The 

broad field of analytical investigation will 

opened t<> his intellectual vision, and he will 

haw made the first steps in that sublime science 

Iheyleadtp w hich discoven the lawi of nature in their n 

general laws. 

secret hiding-places, and follows them, as they 
reach out, in omnipotent power, to control the 
motions of matter through the entire 

occupied spa 



BOOK III. 

UTILITY OF MATHEMATICS. 



CHAPTER I . 

THE UTILITY OF MATHEMATICS CONSIDERED AS A MEANS OF INTELLECTUAL 
TRAINING AND CULTURE. 

§ 316. The first efforts in mathematical sci- First efforts. 
ence are made by the child in the process of 
counting. He counts his fingers, and repeats 
the words one, two, three, four, five, six, seven, CoUDlmgof 

sensible ob- 

eight, nine, ten, until he associates with these jects. 
words the ideas of one or more, and thus ac- 
quires his first notions of number. Hence, the 
idea of number is first presented to the mind by 
means of sensible objects ; but when once clear- 
ly apprehended, the perception of the sensible 
objects fades away, and the mind retains only 
the abstract idea. Thus, the child, after count- Gcneraii». 

tion. 

ing for a time with the aid of his fingers or his 
marbles., dispenses with these cumbrous helps, and 



294 UTILITY' O * M A T H i: M A T ICS. [HOOK 111. 

Abstraction, employs only the abstract ideas, which his mind 
embraces with clearness and uses with facility. 

Analytical § 317. Ill the fll'St Stages Of the analytical 

methods, where the quantities considered are 

' Biibk represented by the letters of the alphabet, - 

tot Bible objects again lend their aid to enable the 
mind to gain exact and distinct ideas of the 
things considered ; but no sooner are these id 

obtained than the mind loses sight of the tilings 

themselves, and o entirely through the 

instrumentality of symbols. 

Qmmetrj. J 318. So, also, in Geometry. The right line 

may first be presented to the mind, as a black 

ii-i impim- mark on paper, or a chalk mark on a bli 

4bte objects, board, to impress the geometrical definition, that 

11 A straight line does not change it- di 
between any two of its points." When this 
definition is clearly apprehended, the mind n< • 
no further aid from the eye, for the u 
forever imprinted. 

a piano. §319. The idea of a plane surface may be 

Definition: impressed by exhibiting the surface of a polished 

mirror; and thus the mind may be aided in 

■owiQustj* apprehending the definition, that •• a plane Bur- 
ied. , . ...... . . . 

face IS one m which, it any two points be taken 



chap, i.] 



QUA N T.TY- 



-SPACE. 



295 



the straight line which joins them will lie wholly 
in the surface/' But when the definition is 
understood, the mind requires no sensible object 
to aid its conception. The ideal alone fills the 
mind, and the image lives there without any 
connection with sensible objects. 



Ite true 
conception. 



§ 320. Space is indefinite extension, in which 
all bodies are situated. A solid or body is any 
portion of space embracing the three dimensions 
of length, breadth, and thickness. To give to 
the mind the true conception of a solid, the aid 
of the eve may at first be necessary ; but the 
idea being once impressed, that a solid, in a 
strictly mathematical sense, means only a por- 
tion of space, and has no reference to the mat- 
ter with which the space may be filled, the mind 
turns away from the material object, and dwells 
alone on the ideal. 



Space, 
Solid: 



How con- 
ceived. 



What ti 
really is. 



§ 321. Although quantity, in its general sense, Quantity: 
is the subject of mathematical inquiry, yet the 
anguage of mathematics is so constructed, that Language: 
the investigations are pursued without the slight- 
est reference to quantity as a material substance. 
We have seen that a system of symbols, by 
which quantities may be represented, has beei 
adopted., forming a language for the express^ 



How con- 
structed. 



Synibota : 



296 UTILITY OF MATHEMATICS. [BOOK III. 

of ideas entirely disconnected from material ob- 
jects, and yet capable of ex : and repre- 
Nmireof senting such objects. This symbolical language, 

the lan- 

gio£e: at once copious and exact, not only enables us 

to express our known thoughts, in every depart- 

whatitac- raient of mathematical science, but is a potent 

complishes. ... .... , . 

moans ol pushing our inquiries into unexplored 

,i<>ns, and conducting the mind with certainty 
tQ new and valuable truth 

AdfHii^M § 322. The nature of thai culture, which the 

of mend )n j 11( | ull( i rr ,_r ( , rs |, v being trained in the use of 
.•in exact language, in which the i tion be- 

tween the sign and the thing signified is unmis- 
takable, has been well set forth by a living 
author, greatly distinguished for his scientific 
attainments.* Of the pure sciences, i 

HerschH> "Their objects are so definite, and our no- 
tions of them SO distinct, that we can re;: 
about them with an assurance that the words and 
pis of our reasonings are full and true repre- 
sentatives of the things signified; and. COI 

Exact ian- quently, that when we use language or signs in 

guago pre- . . , , . . 

▼onto error, argument, we neither by their use introduce 
extraneous notions, nor exclude any part of the 
case before us from consideration. For exam- 

* Sir John Herschel, Discourse on the study of Natural 
Fhilosophy. 



CHAP. I.] EXACT TERMS, 297 

pie : the words space, square, circle, a hundred, Mathematics 

i • i • i terms exact 

&c, convey to the mind notions so complete 
in themselves, and so distinct from every thing 
else, that we are sure when we use them we 
know and have in view the whole of our own 
meaning. It is widely different with words ex- Different in 
pressing natural objects and mixed relations. other termg 
Take, for instance, Iron. Different persons at- 
tach very different ideas to this w T ord. One who 
has never heard of magnetism has a widely dif- 
ferent notion of iron from one in the contrary 
predicament. The vulgar who regard this metal How iron is 

regarded by 

as incombustible, and the chemist, who sees it the chemist 
burn w r ith the utmost fury, and who has other 
reasons for regarding it as one of the most com- 
bustible bodies in nature; the poet, who uses The poet- 
it as an emblem of rigidity ; and the smith and 
engineer, in whose hands it is plastic, and mould- 
ed like wax into every form ; the jailer, who prizes The jailer: 
it as an obstruction, and the electrician, who The elect™ 
sees in it only a channel of open communication 
by which that most impassable of obstacles, the 
air, may be traversed by his imprisoned fluid, — 
have all different, and all imperfect notions of 
the same word. The meaning of such a term Final Mus- 
is like the rainbow — everybody sees a different 
one, and all maintain it to be the same." 

" It is, in fact, in this double or incomplete 



298 UTILITY OF MATHEMATICS. [bOOKIII. 



incomplete sense of words, that we must look for the origin 
JOf of a very large portion of the errors into which 



source ( 
error. 



we fall. Xow, the study of the abstract sciences, 
Mathematics such as Arithmetic, Geometry, Algebra, &c, 
•ucherrort. while they aflbrd scope for the exercise of rea- 
soning about objects that are, or, at least, may 
be conceived to be, externa] to us; yet, b 
five from these sources of error and mistake, 
Require* • accustom us to the strict use of language 

Btrid nee of . . ( , 

'—npnfR ;m instrument 01 reason, and by familiarizing us 
in our progress towards truth, to walk uprightly 
and straightforward, on firm ground, give us 
that proper and dignified carriage of mind which 
K.v.iits. could never be acquired by having always to 
pick our steps among obstructions and 1< 
fragments, or to steady them in the reeling tem- 
pests of conflicting meanings.' 1 

Two, ways of § 323. Mr. Locke lays down two ways of in- 
acquiring . ' 

knowledge, creasing our knowledge : 

1st. "Clear and distinct ideas with settled 

names ; and, 

2d. "The finding of those which show their 

agreement or disagreement ;"' that is, the search- 
ing out of new ideas which result from the com- 
bination of those that are known. 
First. In regard to the first of these ways. Mr. Locke 

says : M The first is to get and settle in our minds 



CHAP. I.] INCREASING KNOWLEDGE. 299 

determined ideas of those things, whereof we ideas of 
have general or specific names ; at least, of so b^L^ct. 
many of them as w r e would consider and im- 
prove our knowledge in, or reason about." # * # 
" For, it being evident, that our knowledge can- 
not exceed our ideas, as far as they are either im- Reason, 
perfect, confused, or obscure, w r e cannot expect 
to have certain, perfect, or clear knowledge. 5 ' 

§324. Now, the ideas which make up our why it is 
knowledge of mathematical science, fulfil ex- mat ica. 
actly these requirements. They are all im- 
pressed on the mind by a fixed, definite, and 
certain language, and the mind embraces them 
as so many images or pictures, clear and dis- 
tinct in their outlines, with names which sug- 
gest at once their characteristics and properties. 

§ 325. In the second method of increasing second, 
our knowledge, pointed out by Mr. Locke, math- 
ematical science offers the most ample and the why matho 
surest means. The reasonings are all based on ™* 1CS0 

© the surest 

self-evident truths, and are conducted bv means me a ns - 
of the most striking relations between the known 
and the unknown. The things reasoned about, 
and the methods of reasoning, are so clearly 
apprehended, that the mind never hesitates or 
doubts. It comprehends, or it does not compre- 






300 UTILITY OF MATHEMATICS. [boOKIII. 

hend, and the line which separates the known 
characters from the unknown, is always well defined. These 

tic? of the 

reasoning, characteristics give to this system of reasoning 
itsadyan- a superiority over every other, arising, not from 
any difference in the logic, but from a difference 
in the things to which the logic is applied. Ob- 
servation may deceive, experiment may tail, and 
Detnonstra- experience prove treacherous, but demonstration 

tion certain. 

never. 

Mathematics i- It it be true, then, that mathemat ics inciude 
Include la ... 

certain t>8- a penecl system ol reasoning, whose prem 

are self-evident, and who-- conclusions are irre- 

;il>ie. can iliere be ,*iny branch of • oj 

knowledge better adapted to the improvement 

of the understanding? I; is in this capacity! 
Anmijunct as a strong and Datura] adjunct and instrument 

:iih1 in<tru- i 1 • • 1 1 ,• i 

meat of rea- °J reason, that this Science becomes the lit sub- 
ject of education with all conditions of s 
whatever may be their ultimate pursuits. Most 

sciences, as, indeed, mo>t branches of knowled 
address themselves to some particular taste, <>r 
subsequent avocation; but this, while it is be- 
fore all, as a useful attainment, especially adapts 
itself to the cultivation and improvement of the 
and necessa- thinking faculty, and is alike necessary to ail 
° who would be governed by reason, or live for 

usefulness."* 

* Mansfield's Discourse on the Mathematics. 



CHAP. I.] REASONS. 301 



§ 326. The following, among other consider- consijera 
ations, may serve to point out and illustrate the ^ ° e of e 
value of mathematical studies, as a means of mathematica ' 
mental improvement and development. 

1. We readily conceive and clearly appre- first. 

They give 

hend the things of which the science treats ; c iear "concep. 
they being thing? simple in themselves and read- t l °° s ° 
fly presented to the mind by plain and familiar 
language. For example : the idea of number, of 
one or more, is among the first ideas implanted Example, 
in the mind ; and the child who counts his fin- 
gers or his marbles, understands the art of num- 
bering them as perfectly as he can know any 
thing. So, likewise, when he learns the definition They estab- 
ot a straight line, oi a triangle, oi a square, oi relations be- 
a circle, or of a parallelogram, he conceives the tv \ een defi , nl 

7 l o tions and 

idea of each perfectly, and the name and the thing *- 
image are inseparably connected. These ideas, 
so distinct and satisfactory, are expressed in the 
simplest and fewest terms, and may, if necessary, 
be illustrated by the aid of sensible objects. 

2. The words employed in the definitions second. 

. Words are 

are always used in the same sense — each ex- •wysused 
pressing at all times the same idea ; so tha\ s^*** 
w T hen a definition is apprehended, the concep 
tion of the thing, U hose name is defined, is per- 
feet in the mind. 

There is, therefore, no doubt cr ambiguity 



302 UTILITY OF MATIIE M ATKS. [BOOK III. 



Hence, it is either in the language, or in regard to what is 

certain 

affirmed or denied of the things spoken of; but 

all is certainty, both in the language employed 

and in the ideas which it expresses. 

Third. 3. The science of mathematics employs no 

no definition definition which may Dot be clearly eompre- 

" . . *T **f hended — lavs down no axioms not universi 

e\ ident and 

clear. tm(l , lll( | j () which the mind, by the very 1 

of its nature, readily assents ; and because, also, 
in the process erf the reasoning, no principle or 
truth is taken for granted, but every link in 
Thecoma- the chain of the argument is immediately con- 
nected with a definition or axiom, or with some 

principle previously established. 

Fourth. 4. The <>r<]cr established in presenting the 

The order . . . , . . 

strengthens subject to the mind, aids the memory at the 

different fac- 
ulties 



same time that it strengthens and improves the 
reasoning powers. For example: first, there 
now ideas are the definitions of tin 4 names of the things 
an ^£"*' w hich are the subjects of the reasoning; then 
the axioms, or self-evident truths, which, to- 
gether with the definitions, form the basis of the 
science. From these the simplest propositions 
now the de- are deduced, and then follow others of greater 
dnctionsfoi- difficuIty . the whole connected together by rig- 
orous logic — each part receiving strength and 
light from all the others. Whence, it follows, 
that any proposition may be traced to first prin- 



CHAP. I.] 



SYNTHESIS ANALYSIS. 



303 



ciples ; its dependence upon and connection Propositions 
with those principles made obvious ; and its truth the ir source*. 
established by certain and infallible argument. 

5. The demonstrative argument of mathe- Fifth - 

Argument 

matics produces the most certain knowledge the most 

r ' 'ii t i certain. 

of which the mind is susceptible. It estab- 
lishes truth so clearly, that none can doubt or 
deny. For, if the premises are certain — that is, Reason*, 
such that all minds admit their truth without 
hesitation or doubt, and if the method of draw- 
ing the conclusions be lawful — that is, in accord- 
ance with the infallible rules of logic, the infer- 
ences must also be true. Truths thus established 
may be relied on for their verity ; and the knowl- such knowi- 
edge thus gained may well be denominated 



Science. 



Two sys- 
tems: 



$ 327. There are, as we have seen, in mathe- 
matics, two systems of investigation quite differ- 
ent from each other : the Synthetical and the synthesis, 
Analytical ; the synthetical beginning with the Anal >' sifl ' 
definitions and axioms, and terminating in the 
highest truth reached by Geometry. 

" This science presents the very method by g riiynffjrJ 
which the human mind, in its progress from 
childhood to age, develops its faculties. What 
first meets the observation of a child ? Upon Firgt notiona 
what are his earliest investigations employed ? 



304 UTILITY OF MATHEMATICS. [BOOK III 



what is first Next to color, which exists only to the sight, 

observed. . . . , , 

figure, extension, dimension, are the first objects 
which he meets, and the first which he examines. 
He ascertains and acknowledges their existence ; 
then he perceives plurality, and begins to enu- 
rogressof merate ; finally he begins to draw conclusions 
from the parts to the whole, and makes a law 
from the individual to the species. Thus, he 
has obtained figure, extension, dimension, enu- 
meration, and generalization. This is the teach- 
ing of nature: and hence, when this proc 
Processde- heroines enil M >died in a perfect system, as it is 

veloped in . . . 

the system of 1U Geometry, that system becomes the easiest 
Geometry, ^j m W j natural means of strengthening the 

mind in its early progress through the fields of 

knowledge.' 1 

Firstnr.-rs- "Long after the child has thus begun to gen- 
An SLu, eralize and deduce laws, lie notices objects and 
events, whose exterior relations afford no con- 
clusion upon the subject of his contemplation. 
Machinery is in motion — effects are produced. 

its method. He is surprised; examines and inquires. He 
reasons backward from elfect to cause. This is 
Analysis, the metaphysics of mathematics ; and 
what the through all its varieties — in Arithmetic — in Alge- 
bra — and in the Differential and Integral Calcu- 
lus, it furnishes a grand armory of weapons for 
acute philosophical investigation. But analysis 



science is : 



chap, i.] bacon's opinion, 305 

advances one step further by its peculiar nota- what it does 
tion; it exercises, in the highest degree, the fac- 
ulty of abstraction, which, whether morally or 
intellectually considered, is always connected 
with the loftiest efforts of the mind. Thus this 
science comes in to assist the faculties in their 
progress to the ultimate stages of reasoning; 
and the more these analytical processes are cul- what it 

finally ac- 
tivated, the more the mind looks in upon itself, compiishes. 

estimates justly and directs rightly those vast 

powers which are to buoy it up in an eternity 

of future being."* 

§ 328. To the quotations, which have already 
been so ample, we will add but two more. 

" In the mathematics, I can report no defi- Bacon's 

, , , rr. opinion of 

cience, except it be that men do not sum- mat hematica. 
ciently understand the excellent use of the pure 
mathematics, in that they do remedy and cure 
many defects in the wit and faculties intellectual. 
For, if the wit be too dull, they sharpen it ; if 
too wandering, they fix it ; if too inherent in the 
sense, they abstract it."f Again : 

" Mathematics serve to inure and corroborate How the 

i-i tt • i study of 

the mind to a constant diligence in studv, to 



* MansfielcTs Discourses on Mathematics 
f Lord Bacon. 



20 



306 UTILITY OF MATHEMATICS. [bOOKIII. 

mathematics undergo the trouble of an attentive meditation, 

mincL and cheerfully contend with such difficulties as 

lie in the way. They wholly deliver us from 

credulous simplicity, most strongly fortify us 

against the vanity of skepticism, effectually re- 

ita influences, strain us from a rash presumption, most easily 
incline us to due assent, perfectly subjugate us 
to the government and weight of reason, and 
inspire us with resolution to wrestle Bgainsl the 
injurious tyranny of false prejudie. 

Nowthcyare " If the fancy he unstable and fluctuating, it 

exerted. 

is, as it were, poised by this ballast, and steadied 

by this anchor; it' the wit be blunt, it is sharp- 
ened by this whetstone; if it be luxurant, it is 

pruned by this knife : if it he headst nag, it IS 

retrained by this bridle ; and if it be dull, it is 

roused by this spur."* 

§ 3*29. Mathematics, in all its branches, is, in 

fact, a science of ideas alone,, unmixed with mat- 
Mnthematics tor or material things; and hence, is properly 
termed a Pure Science. It is, indeed, a fairy 
land of the pure ideal, through which the mind 
is conducted by conventional symbols, as thought 
is conveyed along wires constructed by the 
hand of man. 

* Dr. Barrow 



euce. 



CHAP. I.] CONCLUSION 307 

§ 330. In conclusion, therefore, we may claim what may 
for the study of Mathematics, that it impresses claimed tor 
the mind with clear and distinct ideas ; culti- ma ema 1CS 
vates habits of close and accurate discrimina- 
tion ; gives, in an eminent degree, the power 
of abstraction ; sharpens and strengthens all the 
faculties, and develops, to their highest range, 
the reasoning powers. The tendency of this its tendency, 
study is to raise the mind from the servile habit 
of imitation to the dignity of self-reliance and 
self-action. It arms it with the inherent ener- 
gies of its own elastic nature, and urges it out The reason* 
on the great ocean of thought, to make new 
discoveries, aid enlarge the boundaries of men- 
tal effort. 



308 UTILITY OF MATHEMATICS. | BOOK III 



CHAPTER II. 

TUE LTlUiY Of K&THXKATICB MDG \ki>i:i> as a mi. an- Of AcgnuiNQ 

KNoWI.I .l.r.l. H.V cMAN nil I.( >S< >1MI V. 

Mathematics: § 331. Iii the preceding chapter, we consid- 
ered the effects of mathematical studies on the 
mind, merely as a means of discipline and train- 
How consid- ing. We regarded the study in a single point 

ered hereto- , . , 

fore: oi view, viz. as tlic drill-master of the intel- 
lectual faculties — the power best adapted to 
bring them all into order — to impart strength, 
and to give to them organization. In the 
How now present chapter we shall consider the study un- 
der a more enlarged aspect — as furnishing to 
man the keys of hidden and precious knowl- 
edge, and as opening to his mind the whole 
volume of nature. 



Material § 332. The material universe, which is spread 

Universe. , , . 

out before us, is the first object oi our rational 



considered. 



CHAP. II.] MATERIAL UNIVERSE. 309 

regards. Material things are the first with which 

we have to do. The child plays with his toys Elements of 

know led "0 

in the nursery, paddles in the limpid water, 
twirls his top, and strikes with the hammer. 
At a maturer age a higher class of ideas are 
embraced. The earth is surveyed, teeming with 
its products, and filled with life. Man looks 
around him with wondering and delighted eyes, obtained by 

observation. 

lhe earth he stands upon appears to be made 
of firm soil and liquid waters. The land is 
broken into an irregular surface by abrupt hills 
and frowning mountains. The rivers pursue 
their courses through the valleys, without any course of 

nature : 

apparent cause, and finally seem to lose them- 
selves in their own expansion. He notes the 
return of day and night, at regular intervals, 
turns his eyes to the starry heavens, and in- 
quires how far those sentinels of the night may 
be from the world they look down upon. He 
is yet to learn that all is governed by general Governed 
laws imparted by the fiat of Him who created ' i aws: 
all things ; that matter, in all its forms, is sub- 
ject to those laws ; and that man possesses the Man P os- 
capacity to investigate, develop, and understand fa ^^ to L 
them. It is of the essence of law that it in- yest f ate ■** 

understand 

eludes all possible contingencies, and insures tne ™ 
implicit obedience ; and such are the laws of 
nature. 



310 UTILITY OF MATHEMATICS. [llOOK JII 



§ 333. To the man of chance, nothing is more 
mysterious than the developments of science. 

uniformity: He does not see how so great a uniformity can 
variety : jonsist with the infinite variety which pervades 
every department of nature. While no two 
individuals of a species are exactly alike, the 
resemblance and conformity are so close, that 
the naturalist, from the examination of a sin- 
gle hone, finds no difficulty in determining the 
species, size, and structure of the animal. So. 

Theynppear a ] SOj j u the vegetable and mineral kingdoms: 

in all things. 

all the structures of growth or formation, al- 
though infinitely varied, are yet conformable to 

like general laws, 
science ne- This wonderful mechanism, displayed in the 
thedevei- structure of animals, was but imperfectly und 

° P Taw t0f stooc1 ' until touclie(i t>y the magic wand of sci- 
ence. Then, a general law was found to per- 
vade the whole. Every bone is of that length 
Wnatscienco and diameter best adapted to its use ; every 
muscle is inserted at the right point, and works 
about the right centre ; the feathers of every 
bird are shaped in the right form, and the curve* 
in which they cleave the air are best adapted 

what may to velocity. It is demonstrable, that in every 

bo demon- 

Btrated. case, and in all the variety of forms in which 
forces are applied, either to increase power or 
gain velocity, the very best means have been 



CHAP. II.] PHILOSOPHY OF BACON. 311 



adopted to produce the desired result. And why why it is so. 
should it not be so, since they are employed 
by the all- wise Architect ? 

§ 334. It is in the investigations of the laws Applications 
of nature that mathematics finds its widest Mathematics, 
range and its most striking applications. 

Experience, aided by observation and enlight- 
ened by experiment, is the recognised fountain Bacon's 

Philosophy. 

of all knowledge of nature. On this foundation 
Bacon rested his Philosophy. He saw T that the 
Deductive process of Aristotle, in w T hich the 
conclusions do not reach beyond the premises, Aristotle's: 
was not progressive. It might, indeed, improve 
the reasoning powers, cultivate habits of nice 
discrimination, and give great proficiency in 
verbal dialectics ; but the basis was too narrow 
for that expansive philosophy, which was to its defects, 
unfold and harmonize all the laws of nature. 
Hence, he suggested a careful examination of what Bacon 
nature in every department, and laid the foun- suggestet * 
dations of a new philosophy. Nature was to 
be interrogated by experiment, observation was 
to note the results, and gather the facts into the 
storehouse of knowledge. Facts, so obtained, The means tc 
were subjected to analysis and collation, and 
general law T s inferred from such classification by 



312 UTILITY OF MATHEMATICS. [BOOK III, 



Bacon's a reasoning process called Induction. Hence, 
inductee, ^e system of Bacon is said to be Inductive. 



New Phiioso- § 335. This new philosophy gave a startling 

p y ' impulse to the human mind. Its subject was 

Nature — material and immaterial ; its object, the 

discovery and analysis of those general laws 

what u did. which pervade, regulate, and impart uniformity 
to all things ; its processes, experience, experi- 
ment, and observation for the ascertainment of 
itanature. facts ; analysis and comparison for their classifi- 
cation; and reasoning, for the establishment of 

What aided general laws. But the work would have ! 

incomplete without the aid of deductive science. 
General laws deduced from many separate cas 
whatu by Induction, needed additional proof; tor, they 
might have been inferred from resemblances too 
slight, or coincidences too few. Mathematical 
science affords such proofs. 

The truths of §336. Regarding general laws, established by 
induction: i nc } uct i on? as fundamental truths, expressing these 
by means of the analytical formulas, and then 
operating on these formulas by the known pro- 
How verified cesses of mathematical science, we are enabled, 

by Analysis. ... . 

not only to verity the truths of induction, but 
often to establish new truths, which were hidden 
from experiment and observation. As the in- 



CHAl V] EXPERIMENTAL SCIENCE. 313 

ductive pruCi*^ iT^cy : nvolve error, while the 
deductive cannot, there are weighty scientific 
reasons, for giving to every science as much 
of the character of a Deductive Science as pos- 
sible. Every science, therefore, should be con- As far as 

, . ! | ~ i • l mi possible, all 

structed with the tew T est and simplest possible sciences 
inductions. These should be made the basis of **f^? 

made Deduc- 

deductive processes, by which every truth, how- tive - 
ever complex, should be proved, even if we 
chose to verify the same by induction, based on 
specific experiments. 

§ 337. Every branch of Natural Philosophy Natural pli. 

. . . ' losophy waa 

was originally experimental ; each generahza- experimen- 
tion rested on a special induction, and was de- 
rived from its own distinct set of observations 
and experiments. From being sciences of pure 
experiment, as the phrase is, or, to speak more 
correctly, sciences in which the reasonings con- is now 

deductive. 

sist of no more than one step, and that a step 

of induction ; all these sciences have become, 

to some extent, and some of them in nearly their 

whole extent, sciences of pure reasoning : thus, 

Tiultitudes of truths, already known by indue- 

:ion, from as many different sets of experiments, 

have come to be exhibited as deductions, or co- Mathemati- 

rollaries from inductive propositions of a simpler calor 

and more universal character. Thus, mechan- 



814 UTILITY OF MATHEMATICS. [bOOKIII 



Deductive ics, hydrostatics, optics, and acoustics, have 

Sciences i 

successively been rendered mathematical ; and 
astronomy was brought by Newton within the 
Jaws of general mechanics. 
Their advan- The substitution of this circuitous mode of 

tiiges : 

proceeding for a process apparently much easier 

and more natural, is held, and justly too, to be 

the greatest triumph in the investigation of nature. 

They rest on But, it is necessary to remark, that although, by 

Inductions. . 

this progressive transformation, all sciences tend 
to become more and more deductive, they are 
not, therefore, the less inductive; far, every step 

in the deduction rests upon an antecedent in- 
Sciencesde- duction. The opposition is. perhaps, not 

ductive or ex- 
perimental, much between the terms Deductive and Induc- 
tive as between Deductive and Experimental. 



Ex erimen- § 3 *^' ^ science is experimental,, in propor- 

tal Science: j-j on as evcrv new case, which presents any pe- 

culiar features, stands in need of a new set of 
observations and experiments, and a fresh in- 
duction. It is deductive, in proportion as it can 
Whende- draw conclusions, respecting cases of a new 
ductive. kind, by processes which bring these cases un- 
der old inductions, or show them to possess 
known marks of certain attributes. 

§ 339. We can now, therefore, perceive, what 






CHAP. II.] DEDUCTIVE SCIENCES. 315 

is the generic distinction between sciences that Difference 
can be made deductive and those which must, ductiveond 
as yet, remain experimental. The difference Ex ^ riraental 

J L e _-nce3. 

consists in our having been able, or not yet 
able, to draw from first inductions as from a 
general law, a series of connected and depend 
ent truths. When this can be done, the de 
ductive process can be applied, and the sci- 
ence becomes deductive. For example : when Mm&h 
Newton, by observing and comparing the mo 
tions of several of the heavenly bodies, discov 
ered that all the motions, whether regular 01 Exampi . 
apparently anomalous, of all the observed bodies 
of the Solar System, conformed to the law of 
moving around a common centre, urged by a 
centripetal force, varying directly as the mass, 
and inversely as the square of the distance from 
the centre, he inferred the existence of such a what New. 
law for all the bodies of the system, and then de- 
monstrated, by the aid of mathematics, that no 
other law could produce the motions. This is what he 
the greatest example which has yet occurred of prmed * 
the transformation, at one stroke, of a science 
which was in a great degree purely experimen- 
tal, into a deductive science. 

§ 340. How far the study of mathematics pre- Stud v f 
pares the mind for such contemplations and mathematics 



316 UTILITY OF MATHEMATICS. [BOOK III, 



prepares the such knowledge, is well set forth by an old wri- 
ter, himself a distinguished mathematician. He 
says : 

Dr. Barrow's w The steps are guided by no lamp more clear- 
opinion. 

]y through the dark mazes of nature, by no thread 
more surely through the infinite turnings of the 
labyrinth of philosophy ; nor lastly, is the bottom 
of truth sounded more happily by any other line, 
iiow I vriD not mention with how plentiful a stock 

mathematics 

iWniahthe of know ledge the mind is furnished from th< 

with what wholesome food it is nourished, anc 
what sincere pleasure it enjoys. But if I speak 
further, I shall neither be the only person noi 
the first, who affirms it. that while the mind is 

Abstract abstracted, and elevated from sensible matter, 

mid elevate ..... - . , , 

it: distinctly views pure forms, conceives the beau- 
ty of ideas, and investigates the harmony of pro- 
portions, the manners themselves are sensibly 
corrected and improved, the affections composed 
and rectified, the fancy calmed and settled, and 
the understanding raised and excited to more 
conflrmedby divine contemplations : all of which I might de- 

philosophers. 

fend by the authority and confirm by the suf- 
frages of the greatest philosophers."* 

Horschei'a § 341. Sir John Herschel, in his Introduction 

* Dr. Barrow. 



CHAP. II.] ASTRONOMY \V I T HO UT MATHEMATICS. 317 

to his admirable Treatise on Astronomy, very opinions. 

justly remarks, that, 

" Admission to its sanctuary [the science of Mathemat- 
ical science, 

Astronomy], and to the privileges and feelings indispensa- 
of a votary, is only to be gained by one means — knowledge of 
sound and sufficient knowledge of mathematics, AstroDOm >- 
the great instrument of all exact inquiry, with- 
out which no man can ever make such advances 
in this or any other of the higher departments 
of science as can entitle him to form an inde- 
pendent opinion on any subject of discussion 
within their range. 

" It is not without an effort that those who informa- 

..... . tion cannot 

possess this knowledge can communicate on be given 



such subjects with those who do not, and adapt 



to such as 
have no 

their language and their illustrations to the ne- mathemalics; 
cessities of such an intercourse. Propositions 
which to the one are almost identical, are the- 
orems of import and difficulty to the other ; nor 
is their evidence presented in the same way to 
the mind of each. In treating such proposi- Except 
tions, under such circumstances, the appeal has b y r ous m™h^ 
to be made, not to the pure and abstract reason, ods * 
but to the sense of analogy — to practice and 
experience : principles and modes of action have 
to be established, not by direct argument from 
acknowledged axioms, but by continually refer- 
ring to the sources from which the axioms them- Reasons 



318 UTILITY OF MATHEMATICS. [BOOK III. 



Must begin selves have been drawn, viz. examples ; that is to 

with the sim- i i • • r 1 ] i 1 1 • i 

piesteie- sa y> ®y bringing iorward and dwelling on simple 
nu-nts: an( j f am ili a r instances in which the same prin- 
ciples and the same or similar modes of action 
take place ; thus erecting, as it were, in each 
particular case, a separate induction, and con- 
structing at each step a little body of science to 

illustration meet its exigencies. The difference is that of 

of the differ- 
ence be- pioneering a road through an untraversed coun- 

structionby t! T' an( l advancing at ease along a broad and 
scientific and b ea t en highway ; that is to sav, if we are dcter- 

unscientiiic ° J 

methods, mined to make ourselves distinctly understood, 
and will appeal to reason at all." Again: 
Mathcmntics " A certain moderate degree of acquaintance 

uecessaryto ^.j^ a } )stract sc i cn(V j s highly dt'Sirablo tO CVCl'V 
physics : ° J * 

one who would make any considerable progress 
in physics. As the universe exists in time and 
place ; and as motion, velocity, quantity, num- 
ber, and order, are main elements of our knowl- 
edge of external things and their changes, an 
acquaintance with these, abstractedly consid- 
whyitisso ered (that is to say, independent of any consid- 
necKatry. erat j on Q f p ar ticular things moved, measured, 
counted, or arranged), must evidently be a use- 
ful preparation for the more complex study of 



* Sir John Herschel on the study of Natural Philosophy, 



CHAP. II.] ASTRONOMY. 319 



§ 342. If we consider the department of ehem- Necessary in 

, . . , 'ii chemistry. 

istry, — -which analyzes matter, examines the ele- 
ments of which it is composed, develops the laws 
which unite these elements, and also the agencies 
which will separate and reunite them, — we shall 
find that no intelligent and philosophical analysis 
can be made without the aid of mathematics. 

§ 343. The mechanism of the physical uni- Laws long 

unknown. 

verse, and the laws which govern and regulate 
its motions, were long unknown. As late as the 
17th century, Galileo was imprisoned for pro- Gauieo. 
mulgating the theory that the earth revolves on 
its axis; and to escape the fury of persecution, His theory, 
renounced the deductions of science. Now, ev- 
ery student of a college, and every ambitious boy gov k< 
of the academy, may, by the aid of his Algebra 
and Geometry, demonstrate the existence and 
operation of those general laws which enable By what 

. . . , . . , , means de- 

him to trace with certainty the path and mo- monstr ated. 
tions of every body which circles the heavens. 

§ 344. What knowledge is more precious, or value 
more elevating to the mind, than that which ° f scie " tific 

3 knowledge : 

assures us that the solar system, of which the 
sun is the centre, and our earth one of the 
smaller bodies, is governed by the general law 
of gravitation ; that is, that each body is re- 
tained in its orbit by attracting, and being at- 



lOWU 

to all: 



320 UTILITY OF MATHEMATICS. [BOOK lit. 



what tracted by, all the others ? This power of attrac- 

it teaches 

tion, by which matter operates on matter, is th > 
great governing principle of the material work. 
The motion of each body in the heavens de 

The things pends on the forces of attraction of all th. 
others ; hence, to estimate such forces — varyim 
as they do with the quantity of matter in eaci 
body, and inversely as the squares of their dis- 
tances apart — is no easy problem ; yet analy- 

Anaiysis: sis has solved it, and with such certainty, that 

the exact spot in the heavens may be marked 

at which each body will appear at the expiration 

what it has f anv definite period of time. Indeed, a tele- 
done: 

scope may be so arranged, that at the end of 

How a that time either one of the heavenly bodies 
be verified would present itself to the field of view; and 
byexpen- ^ j instrument could remain fixed, though the 
time were a thousand years, the precise mo- 
ment would discover the planet to the eye of the 
observer, and thus attest the certainty of science. 

§ 345. But analysis has done yet more. It 

has not only measured the attractive power of 

Analysis eac h f the heavenly bodies ; determined their 

determines 

balancing distances from a common point and from each 
other; ascertained their specific gravities and 
traced their orbits through the heavens; but 
has also discovered the existence of balancing 



forces. 






CHAP. II. j STABILITY OF THE UNIVERSE. 321 



and conservative forces, evincing the highest Evidence a 
evidence of contrivance and design. 



§ 346. A superficial view of the architecture Architecture 
of the heavens might inspire a doubt of the sta- e ns shows 
bility of the entire system. The mutual action permancncy 
of the bodies on each other produces what is 
called an irregularity in their motions. The 
earth, for example, in her annual course around Example of 

rr - c tne eartn • 

the sun, is affected by the attraction of the 
moon and of all the planets which compose the 
solar system; and these attracting forces appear 
to give an irregularity to her motions. The 
moon in her revolutions around the earth is also ot tin moon, 
influenced by the attraction of the sun, the earth, 
and of all the other planets, and yields to each a 
motion exactly proportionate to the force exert- 
ed ; and the same is equally true of all the bodies or the other 

planets. 

which belong to the system. It was reserved 
for analysis to demonstrate that every supposed 
irregularity of motion is but the consequence of 
a general law ; that every change is constancy, 
and every diversity uniformity. Thus, mathe- Mathematics 

, . . proves thD 

matical science assures us that our system has permanency 
not been abandoned to blind chance, but that oftbes y 3 - 

7 tern. 

a superintending Providence is ever exerted 

through those general laws, which are so minute 

as to govern the motions of the feather as it is 

21 



322 UTILITY OF MATHEMATICS. [BOOK III, 



Generality of wafted along on the passing breeze, and vet so 



Jaws. 



omnipotent as to preserve the stability of worlds. 



§ 347. But analysis goes yet another step. 

That class of wandering bodies, known to us by 

comets: the name of comets, although apparently escaped 

from their own spheres, and straying heedlessly 

Ulmt through illimitable space, have vet been pursued 

prnY.-s in re- j,y the telescope of the observer until sufficient 
gnrd to them« 

data have been obtained to apply the pTOC 

of analysis. This done, a few lines written upon 

paper indicate the precise times of their reap- 
Rraultsitfi- pearance. These results, when first obtained, 

were so striking, and apparently so far beyond 

the reach of science itself, as almost to need 

verification, the verification of experience. At the appointed 

times, however, the COmetfl reappear, and ! 
ence is thus verified by observation. 



Nature § 348. The great temple of* nature is only to 

cannot be in- . . . 

vestigated be opened by the keys ot mathematical science. 

■nJfK^l^ We may perhaps reach the vestibule, and gaze 
with wonder on its gorgeous exterior and its 
exact proportions, but we cannot open the por- 

niustration. tal and explore the apartments unless we use 
the appointed means. Those means are the 
exact sciences, which can only be acquired by 
discipline and severe mental labor. 



;HAP. II.] REsULTS OF SCIENCE. 323 

The precious metals are not scattered pro- sdeuce: 
fusely over the surface of the earth ; they are, 
for wise purposes, buried in its bosom, and can 
be disinterred only by toil and labor. So with 
science : it comes not by inspiration ; it is not 
borne to us on the wings of the wind ; it can ouiy to be 

acquired bj 

neither be extorted by power, nor purchased by sludy: 
wealth ; but is the sure reward of diligent and 
assiduous labor. Is it worth that labor ? What Tt is worth 

study. 

is it not worth ? It has perforated the earth, 

and she has yielded up her treasures ; it has Wh* 

it has done 

guided in safety the bark of commerce over dis- f or the wantt 
tant oceans, and brought to civilized man the 
treasures and choicest products of the remotest 
climes. It has scaled the heavens, and searched 
out the hidden laws which regulate and govern 
the material universe ; it has travelled from 
planet to planet, measuring their magnitudes, sur- 
veying their surfaces, determining their days and 
nights, and the lengths of their seasons. It has 
also pushed its inquiries into regions of space, what 
where it w T as supposed that the mind of the to make us 
Omnipotent never yet had energized, and there ac( i uainte(1 

1 «/ o W1 th the uni- 

located unknown worlds — calculating their di- verse - 
ameters, and their times of revolution. 



§ 349. Mathematical science is a magnetic how 
telegraph, which conducts the mind from orb mathcmaU< * 



324 



UTILITY OF MATHEMATICS. [BOOK III. 



aid the to orb through the entire regions of measured 

mind in its 

inquiries: space. It enables us to weigh, in the balance 
of universal gravitation, the most distant planet 
of the heavens, to measure its diameter, to de- 
termine its times of revolution about a common 
centre and about its own ;i\is. and to claim it 
as a part of our own system. 

Bow they In these far teachings of the mind, the im- 
eotauye it : 

agination has full scope for its highest exerc 
It is not led astray by the false ideal and fed by 
illusive visions, which sometimes tempt reason 
from her throne, but is ever guided by the- tie- 
Maybe duotioiis of science; and its ideal and the real 

reli(,<l im - 1 1 1 /• 1 ] • 1 

are united by the li\ed laws <>i eternal truth. 



Mind § 350. There is that Within us which d« lights 

delights in . . riM . . , . . . 

nrtainty. m Certainty. I he IlllstS ot (loul)t ODSCUl^fi t ho 

mental, as the mists of the morning do the phys- 
ical vision. We love to look at nature through 
a medium perfectly transparent, and to see every 

object in its exact, proportions. The science of 
Wfcj mathematics is that medium through which the 

mathematics 

affoniit. mind may view, and thence understand all the 



parts of the physical universe. It makes man- 
ifest all its laws, discovers its wonderful harmo- 
nies, and displays the wisdom and omnipotence 
of the Creator. 



CHAP, ill.] "practical/' 325 



CHAPTER III. 



THE UTILITY OF MATHEMATICS CONSIDERED AS FURNISHING THOSE RULES Of 
ART WHICH MAKE KNOWLEDGE PRACTICALLY EFFECTIVE. 



§ 351. There is perhaps no word in the Eng- Practical: 
lish language less understood than Practical. Little 

-p. . . , , ■ i « understood. 

by many it is regarded as opposed to theoreti- 
cal. It has become a pert question of our day, its popular 

TIT . . . . iri i i signification. 

" Whether such a branch oi knowledge is prac- 
tical ?" " It" any practical good arises from pur- Qamtmi 

, l rv, 3 t ■ • i r n relating to 

suing such a study r - It it be not lull time stuc iiesand 
that old tomes be permitted to remain untouched 
in the alcoves of the library, and the minds of 
the young fed with the more stimulating food of 
modern progress ?" 



§352. Such inquiries are not to be answered inquiries- 
by a taunt. They must be met as grave ques- How to be 

considered 

tions, and considered and discussed with calm- 
ness. They have possession of the public mind ; Their 

influence. 

they anect the foundations of education ; they 



326 



UTILITY OF MATHEMATICS. 



[book III. 



Their influence and direct the first steps ; they control 



importance. 



the very elements from which must spring the 
systems of public instruction. 



Practical: §353. The term "practical, 7 ' in its common 

common acceptation, that is, in the sense in which it is 
icceptation: ... c r \ 

often used, relers to the acquisition of useful 
knowledge by a short process. It implies a sub- 
stitution of natural sagacity and " mother wit" 
for the results of hard study and laborious effort 
It implies the use of knowledge before its acqui- 
sition ; the substitution of the results of mere 
experiment for the deductions of science, and 
the placing of empiricism above philosophy. 



What if 
implies 



In f his scns«, 
It is opposed 
to progress : 



Conse- 
quences. 



Right 
•Ignificatior 



§ 354. In this view, the practical is adverse 
to sound learning, and directly opposed to real 
progress. If adopted, as a basis of national edu- 
cation, it would shackle tin 4 mind with the iron 
letters of mere routine, and chain it down to 
the drudgery of unimproving labor. Under 
such a system, the people would become imita- 
tors and rule-men. Great and original principles 
would be lost sight of, and the spirit of inves- 
tigation and inquiry would find no field for its 
legitimate exercise. 

But give to " practical" its true and right 
signification, and it becomes a word of the 



CHAP. III.] ILLUSTRATED. 327 

choicest import. In its right sense, it is the best Best means 
means of making the true idea] the actual; that tno^ige! 
is, the best means of carrying into the business 
and practical affairs of life the conceptions and 
deductions of science. All that is truly great 
in the practical, is but the actual of an antece- 
dent ideal. 



§ 355. It is under this view that we now pro- Mathemati- 
cal science : 

pose to consider the practical advantages of 
mathematical science. In the two preceding 
chapters we haye pointed out its yalue as a 
means of mental development, and as affording 
facilities for the acquisition of knowledge. We 
shall now show how intimately it is blended its practical 
with the every -day affairs of life, and point out 
some of the agencies which it exerts in giving 
practical development to the conceptions of the 
mind. 



§ 356. We begin with Arithmetic, as this Arithmetic 

considered 



branch of mathematics enters more or less into 



practically 



all the others. And what shall we say of its 
practical utility ? It is at once an evidence and 
element of civilization. By its aid the child in 
the nursery numbers his toys, the housewife 
keeps her daily accounts, and the merchant sums 
up his daily business. The ten little characters, 



328 UTILITY OF MATHEMATICS. [BOOK III. 

which we call figures, thus perform a very im- 

what figures portant part in human affairs. They are sleepless 
sentinels watching over all the transactions of 
trade and commerce, and making known their 
final results. They superintend the entire busi- 

Their value, ness affairs of the world. Their daily records 
exhibit the results on the stock exchange, and 
of enterprises reaching over distant seas. The 

usedbythe mechanic and artisan express the final results of 
all their calculations in figures. The dimensions 

intending, of buildings, their length, breadth, and height, as 
well as the proportions of their several parts. BUS 

all expressed by figures before the foundation 

Aidscience. stones are laid: and indeed, all the results ^i 

Science are reduced to figures before they can 

be made available in practice. 

§ 357. The rules and practice of all the me- 
chanic arts are but applications of mathematical 

Mathematics science. 'The mason computes the quantity of 

useful in the 

mechanic his materials by the principles of Geometry and 

the rules of Arithmetic. The carpenter frames 
his building, and adjusts all its parts, each to 
the others, by the rules of practical Geometry. 
Examples The millwright computes the pressure of the 
water, and adjusts the driving to the driven 
wheel, by rules evolved from the formulae of 
analvsis. 



CHAP. III.] ILLUSTRATED. 329 

§ 358. Workshops and factories afford marked workshops 
illustrations of the utility and value of practical exhibit ap _ 
science. Here the most difficult problems are potions of 

science. 

resolved, and the power of mind over matter 
exhibited in the most striking light. To the * 

uninstructed eye of a i asual observer, confusion 
appears to reign triumphant. But all the parts Parts ad- 

justed on a 

of that complicated machinery are adjusted to general plan. 

each other, and were indeed so arranged, and 

according to a general plan, before a single 

wheel was formed by the hand of the forger. 

The power necessary to do the entire work was Powei 

first carefully calculated, and then distributed and 

throughout the ramifications of the machinery. dlstnbuted * 

Each part was so arranged as to fulfil its office. 

Every circumference, and band, and cog, has 

its specific duty assigned it. The parts are Parts At in 

i v/v i r their proper 

made at amerent places, alter patterns formed places. 
by the rules of science, and when brought to- 
gether, fit exactly. They are but formed parts 
of an entire whole, over which, at the source 
of power, an ingenious contrivance, called the 
Governor, presides. His function is to regulate Governor- 
the force which shall drive the whole according 
to a uniform speed. He is so intelligent, and 
of such delicate sensibility, that on the slightest its functions, 
increase of velocity, he diminishes the force, and 
adds additional power the moment the speed 



330 UTILITY OF MATHEMATICS. [BOOK III. 

aii is but slackens. All this is the result of mathematical 
•donee calculation. When the curious shall visit tl; 

exhibitions of ingenuity and skill, let them not 
suppose that they are the results of chance and 
experiment. They are the embodiments, by in- 
telligent labor, of the most difficult investigatu 
of mathematical sciem 

§369. Another striking example of the appli- 
cation of the principles of science is found in 

moinnhlf Ae steam-hip. 

In the first place, the formation of her hull, 

Bon On hull 80 as to divide the waters with the least re-ist- 

isr,,nm ' ,K ance. and at the same time reeeive from thmi 

the greatest pressure as they close behind her, 

HcrwMto: MS not an easy problem. Her mastfl are all 

Qow to be set at the proper tingle, and her sails so 

■*■*■* adjusted aa to gain a maximum fofCC. Hut the 

complication of her machinery, unless seen 

through the medium of science, baffles investi- 
gation, and exhibits a startling miracle. The 

burning furnace, the immense boilers, the mi 
Machinery: ivc cylinders, the huge levers, the pipes, the 
lifting and closing valves, and all the nicely- 
adjusted apparatus, appear too intricate to be 
comprehended by the mind at a single glance. 

Tbewboto Yet in all this complication — in all this variety 

constructed ., • , , i i • _ i ^.. 

oi principle and workmanship, science has I 



CHAP. III.] ILLUSTRATED. 331 

erted its power. There is not a cylinder, whose according to 

the principle! 

dimensions were not measured — not a lever, f science: 
whose power was not calculated — nor a valve, 
which does not open and shut at the appointed 
moment. There is not, in all this structure, a From a 
bo.t, or screw, or rod, which was not provided 
for before the great shaft was forged, and which 
does not bear to that shaft its proper proportion. 
And when the workmanship is put to the test, b>- 

what meana 

and the power 01 steam is urging the vessel on navigated, 
her distant voyage, science alone can direct her 
way. 

In the captain's cabin are carefully laid away, 
for daily use, maps and charts of the port which Her charts, 
he leaves, of the ocean he traverses, and of the 
coasts and harbors to which he directs his way. 
On these are marked the results of much scien- Their 
tific labor. The shoals, the channels, the points ° on ^. 
of danger and the places of security, are all in- 
dicated. Near by, hangs the barometer, con- Barometer: 
structed from the most abstruse mathematical 
formulas, to indicate changes in the weight of 
the atmosphere, and admonish him of the ap- 
proaching tempest. On his table lie the sextant, sextant: 
and the tables of Bowditch. These enable him, 
by observations on the heavenly bodies, to mark 
his exact place on the chart, and learn his posi- Their uses. 
tion on the surface of the earth. Thus, practical 



332 UTILITY OF MATHEMATICS. [BOOK III. 



science science, which shaped the keel of the ship to 

guides the 

ship : its proper form, and guided the hand of the me- 
chanic in every workshop, is, under Providence, 
the means of conducting her in safety over the 
ocean. It is, indeed, the cloud by day and the 

What piUar of fire by night. Guiding the bark of 

thus accom- 

pushes, commerce over trackless waters, it brings dis- 
tant lands into proximity, and into political and 
social relations. 

" We have before us an anecdote communi- 

Ulustration. 

cated to us by a naval officer,* distinguished 

for the extent and variety of his attainments, 

which shows how impressive such results may 

cat. Hair* become in practice. He sailed from San Bias, 

voyage. on ^ wcst coas t of Mexico, and after a voyage 

ite length: of eight thousand miles, occupying eighty-nine 

days, arrived off Rio de Janeiro; having in this 

interval passed through the Pacific Ocean, round- 

and ed Cape Horn, and crossed the South Atlantic, 

incidents. . . 

without making any land, or even seeing a single 
sail, with the exception of an American whaler 
off Cape Horn. Arrived within a week's sail 
of Rio, he set seriously about determining, by 
observations lunar observations, the precise line of the ship's 

taken. 

course, and its situation in it, at a determinate 
moment ; and having ascertained this within 

* Captain Basil Hall. 



CHAP. III.] ILLUSTRATED. 333 

from five to ten miles, ran the rest of the way Remarkable 
by those more ready and compendious methods, coma encc ' 
known to navigators, which can be safely em- 
ployed for short trips between one known point 
and another, but which cannot be trusted in long short 

. methods. 

voyages, where the moon is the only sure guide. 

" The rest of the tale, we are enabled, by his 
kindness, to state in his own words : ' We steered Particulars 

s* at ed 

towards Rio de Janeiro for some days after ta- 
king the lunars above described, and having 
arrived within fifteen or twenty miles of the Arrival at 

Rio. 

coast, I hove-to at four in the morning, till the 
day should break, and then bore up : for although 
it was very hazy, we could see before us a couple 
of miles or so. About eight o'clock it became so 
foggy, that I did not like to stand in further, and 
was just bringing the ship to the wind again, be- 
fore sending the people to breakfast, when it sud- 
denly cleared oft', and I had the satisfaction of recovery o/ 
seeing the great Sugar-Loaf Rock, which stands 
on one side of the harbor's mouth, so nearly right 
ahead that we had not to alter our course above 
a point in order to hit the entrance of Rio. This 
was the first land we had seen for three months, First land in 
after crossing so many seas, and being set back- mon ths. 
wards and forwards by innumerable currents 
and foul winds.' The effect on all on board Effect 
might well be conceived to have been electric : 



334 UTILITY OF MATHEMATICS. [BOCK III. 

on the crew, and it is needless to remark how essentially the 
authority of a commanding officer over his crew 
may be strengthened by the occurrence of such 
incidents, indicative of a degree of knowledge 
and consequent power beyond their reach. 

surveying. § 360. A useful application of mathematical 
science is found in the laying out and measure- 
Measure- ment of land. The necessity of such measure- 
ment, and of dividing the surface of the earth 
into portions, gave rise to the science of Geom* 
Ownership: etry. The ownership of land could not be (le- 
How termined without some means of running boun 

determined. . . . 

dary lines, and ascertaining limits. Levelling 
is also connected with this branch of practical 
mathematics. 

By the aid of these two branches of practical 
science, we measure and determine the area or 

contents of contents of ground ; make maps of its surface ; 
measure the heights of hills and mountains ; 
Rivers. find the directions of rivers ; measure their vol- 
umes, and ascertain the rapidity of their cur- 
rents. So certain and exact are the results, that 
entire countries are divided into tracts of con- 
venient size, and the rights of ownership fully 

Ceriainty secured. The rules for mapping, and the con- 

* Sir John Herschel, on the study of Natural Philosophy 



CHAP. III.] ILLUSTRATED. 335 



ventional methods of representing the surface Mapping. 
of ground, the courses of rivers, and the heights 
of mountains, are so well defined, that the nat- 
ural features of a country may be all indicated Features of 

the ground. 

on paper. Thus, the topographical features of 

all the known parts of the £arth may be cor- Their rcpre- 

rectly and vividly impressed on the mind, by a 

map, drawn according to the rules of art, by the 

human hand. 

§ 361. Our own age has been marked by a Railways, 
striking application of science, in the construc- 
tion of railways. Let us contemplate for a mo- The problem 
ment the elements of the problem which is pre- pre 
sented in the enterprise of constructing a railroad 
between two given points. 

In the first place, the route must be carefully Examination 

, . . , . , .,. of their 

examined to ascertain its general practicability. route3> 
The surveyor, with his instruments, then ascer- surveys. 
tains all the levels and grades. The engineer 

examines these results to determine whether the office of the 

r . ••ill engineer. 

power oi steam, in connection with the best 
combination of machinery, w T ill enable him to 
overcome the elevations and descend the decliv- 
ities in safety. He then calculates the curves calculations 

of curye9. 

of the road, the excavations and fillings, the 
cost of the bridges and the tunnels, if there are 
any ; and then adjusts the steam-power to meet 



336 UTILITY OF MATHEMATICS. [BOOK III. 

completion the conditions. In a few months after the enter- 
prise is undertaken, the locomotive, with its long 
train of passenger and freight cars, rushes over 
the tract with a superhuman power, and fulfils 
the office of uniting distant places in commer- 
cial and social relations. 

The striking But that which is most striking in all this, is 
the fact, that before a stump is grubbed, or a 
spade put into the ground, the entire plan of the 
work, having been subjected to careful analysis, 
is fully developed in all its parts. The construc- 

The whole tion is but the actual of that perfect ideal which 

the result of j j j f witllill itself, and whicll CEO 

science. 

spring only from the far-reaching and immuta- 
ble principles of abstract science. 

§ 362. Among the most useful applications of 

practical science, in the present century, is the 

croton introduction of the Croton water into the city 

aqueduct. of jj^ Y^ 

In the Highlands of the Hudson, about fifty 

miles from the city, the gushing springs of the 

sources of mountains indicate the sources of the Croton 

the river. . . r . , 

river, which enters the Hudson a lew miles 
below T Peekskill. At a short distance from the 
Principal mouth, a dam fifty-five feet in height is thrown 
across the river, creating an artificial lake for 
the permanent supply of water. The area of this 



CHAP. III.] ILLUSTRATED. 337 

lake is equal to about four hundred acres. The its area, 
aqueduct commences at the Croton dam, on a Aqueduct. 
line forty feet above the level of the Hudson 
river, and runs, as near as the nature of the 
ground will permit, along the east bank, till it 
reaches its final destination in the reservoirs 
of the city. There are on the line sixteen tun- its tunnels 
nels, varying in length from 160 to 1,263 feet, 
making an aggregate length of 6,841 feet. The 
heights of the ridges above the grade level of the Their 

heights. 

tunnels range from 25 to 75 feet. Twenty-five 
streams are crossed by the aqueduct in West- streams 

crossed 

Chester county, varying from 12 to 70 feet below 
the grade line, and from 25 to 83 feet below the 
top covering of the aqueduct. The Harlem Harlem river 
river is passed at an elevation of 120 feet above 
the surface of the water. The average dimen- 
sions of the interior of the aqueduct, are about 
seven feet in width and eight feet in height. 

The width of the Harlem river, at the point its width, 
where the aqueduct crosses it, is six hundred 
and twenty feet, and the general plan of the 
bridge is as follows : There ire eight arches, Brid ?° * 
each of 80 feet span, and seven smaller arches, 
each of 50 feet span, the whole resting on piers 
and abutments. The length of the bridge is its length : 
1,450 feet. The height of the river piers from 
the lowest foundation is 96 feet. The arches 

22 



338 UTILITY OF MATHEMATICS. [BOOK III. 

?is height: are semi-circular, and the height from the low- 
est foundation of the piers to the top of the 

its width, parapet is 149 feet. The width across, on the 
top, is 21 feet. 

To afford a constant supply of water for dis- 
tribution in the city two large reservoirs have 

Receiving been constructed, called the receiving reservoir 

Reservoir : 

and the distributing reservoir. The surface of 
the receiving reservoir, at the water-line, is equal 

its extent to thirty-one acres. It is divided into two parts 
by a wall running east and west. The depth of 

Depth of water in the northern part is twenty feet, and 

water. 

in the southern part thirty feet. 

Distributing The distributing reservoir is located on the 
highest ground which adjoins the city, known 

its capacity, as Murray Hill. The capacity of this reservoir 
is equal to 20,000,000 of gallons, which is about 
one-seventh that of the receiving reservoir, and 
the depth of water is thirty-six feet. 
Power The full power of science has not yet been 

illustrated. A perfect plan of this majestic 
structure was arranged, or should have been, 
before a stone was shaped, or a pickaxe put into 
the ground. The complete conception, by a 
single mind, of its general plan and minutest 
details, was necessary to its successful prosecu- 

Whatitac- t j on j t was w ithin the range and power of 

eomplished. 

science to have given the form and dimensions 



CHAP. III.] ILLUSTRATED. 339 

of every stone, so that each could have been 
shaped at the quarry. The parts are so con- connec- 
nected by the laws of the geometrical forms, 10ns a ^g the 
that the dimensions and shape of each stone was 
exactly determined by the nature of that portion 
of the structure to which it belonged. 

§ 363. We have presented this outline of the view of the 
Croton aqueduct mainly for the purpose of aqueduct: 
illustrating the power and celebrating the tri- why given, 
umphs of mathematical science. High intel- 
lect, it is true, can alone use the means in a 
work so complicated, and embracing so great 
a variety of intricate details. But genius, even Little ac _ 
of the highest order, could not accomplish, with- 
out continued trial and laborious experiment, 
such an undertaking, unless strengthened and 
guided by the immutable truths of mathematical 
science. 



§ 364. The examination of this work cannot what 
but fill the mind with a proud consciousness of Jj^ *" 
the power and skill of man. The struggling 
brooks of the mountains are collected together — 
accumulated — conducted for forty miles through 
a subterranean channel, to form small lakes in 
the vicinity of a populous city. 

From these sources, by an unseen process, the 



complished 
without 
science. 



340 UTILITY OF MATHEMATICS. [BOOK III. 

pure water is carried to every dwelling in the 

large metropolis. The turning of a faucet de- 

Conse- livers it from a spring at the distance of fifty 

quences 

which have miles, as pure as when it gushes from its granite 
hills. That unseen power of pressure, which 
resides in the fluid as an organic law, exerts its 
force with unceasing and untiring energy. To 
minds enlightened by science, and skill directed 
by its rules, we are indebted for one of the no- 
blest works of the present century. May we 

cbndik.ion. not, therefore, conclude that science is the only 
sure means of giving practical development to 
those great conceptions which confer lasting 
benefits on mankind? "All that is truly great 
in the practical, is but the result of an antece- 
dent ideal.'' 



APPENDIX. 



A COURSE OF MATHEMATICS WHAT IT SHOULD BE. 

§365. A course of mathematics should pre- a course 
sent the outlines of the science, so arranged, ex- Mathematics, 
plained, and illustrated as to indicate all those 
general methods of application, which render it 
effective and useful. This can best be done by 
a series of works embracing all the topics, and 
in which each topic is separately treated. 

§ 366. Such a series should be formed in ac- How it 

cordance with a fixed plan; should adopt and formed. ' 
use the same terms in all the branches ; should 
be written throughout in the same style ; and 

present that entire unity which belongs to the unity of the 
subject itself. 



subject. 



§ 367. The reasonings of mathematics and Reasonings 
the processes >f investigation, are the same in 



342 APPENDIX. 



the same in every branch, and have to be learned but once, 

if the same system be studied throughout. The 

Different different kinds of notation, though somewhat un- 

kinds of no- 
tation, like in the different subjects of the science, are, 

in fact, but dialects of a common language. 



Language § 368. If, then, the language is, or may be 

need be 

learned but made essentially the same in all the branches of 
mathematical science ; and if there is, as has 
been fully shown, no difference in the processes 
of reasoning, wherein lies that difficulty in the 
acquisition of mathematical knowledge which is 
often experienced by students, and whence the 
origin of that opinion that the subject itself is 
dry and difficult ? 



In what 
consists the 
difficulty ? 



A §309. Just in proportion aa a branch of know- 

general law, ... ! • 1 1 1 

if known, ledge is compactly united by a common law, is 

subject Ity. the facility of acquiring that knowledge, if we 

observe the law, and the difficulty of acquiring 

Faculties it, if we pay no attention to the law. The study 
mthematicl °f mathematics demands, at every step, close 
attention, nice discrimination, and certain judg- 
ment. These faculties can only be developed 

How first by culture. They must, like other faculties, pass 
through the states of infancy, growth, and ma- 
turity. They must be first exercised on sensible 
and simple objects ; then on elementary ab- 



APPExNDIX. 343 



stract ideas; and finally, on generalizations and onwhat 
the higher combinations of thought in the pure cise(L 
ideal. 



§ 370. Have educators fully realized that the Arithmetic 

th:i most im- 

first lessons in numbers impress the first elements portant 
of mathematical science ? that the first con- 
nections of thought which are there formed be- 
come the first threads of that intellectual warp 
which gives tone and strength to the mind ? 
Have they yet realized that every process is, or ah the 
should be, like the stone of an arch, formed to nec te<i. 
fill, in the entire structure, the exact place for 
which it is designed ? and that the unity, beauty, 
and strength of the whole depend on the adapta- 
tion of the parts to each other ? Have they 
sufficiently reflected on the confusion which must Necessity 
arise from attempting to put together and har- ^ p^ 
monize different parts of discordant systems ? 
to blend portions that are fragmentary, and to 
unite into a placid and tranquil stream trains of 
thought which have not a common source ? 

§ 371. Some have supposed that Arithmetic 
may be well taught and learned witl out the aid 
of a text-book ; or, if studied from a book, that a text-book 
the teacher may advantageously substitute his 
own methods for those of the author, inasmuch 



344 APPENDIX. 



tobefoi- as such substitution is calculated to widen the 
field of investigation, and excite the mind of the 
pupil to new inquiries. 

Reasons. Admitting that every teacher of reasonable 

intelligence, will discover methods of communi- 
cating instruction better adapted to the peculiar- 
ities of his own mind, than all the methods em- 
Even a bet- ployed by the author he may use ; will it be safe, 

ter method, 

when aubsti- as a general rule, to substitute extemporaneous 
nothurmo- methods for those which have been subjected 
ni *? with * hc to the analysis of science and the tests of expe- 

other parts J x 

of the work. r j cnce p I s it safe to substitute the results of 
known laws for conjectural judgments ? But if 
they are as good, or better even, as isolated pro- 
cesses, will they answer as well, ID their new 
places and connections, as the parts rejected? 

illustration. Will the balance- wheel of a chronometer give 
as steady a motion to a common watch as the 
more simple and less perfect contrivance to 
which all the other parts are adapted ? 

§ 372. If these questions have significance, wt 
one of the have found at least one of the causes that have 

reasons why 

mathematics impeded the advancement of mathematical soi- 
ls difficult. ence ^ Y i z t j ie attempt to unite in the same course 

of instruction fragments of different systems ; 
thus presenting to the mind of the learner the 
same terms different! v defined, and the same 



APPENDIX. 345 



principles differently explained, illustrated, and 
applied. It is mutual relation and connection connects 

very impor 

which bring sets of facts under general laws ; it tant. 
is mutual relation and connection of ideas which 
form a process of science ; it is the mutual con- 
nection and relation of such processes which 
constitute science itself. 



§ 373. I would by no means be understood as A teacher 
expressing the opinion that a student or teacher should T& f 

1 ° l many books, 

of mathematics should limit his researches to a and teach one 

system. 

single author ; for, he must necessarily read and 
study many. I speak of the pupil alone, who 
must be taught one method at a time, and taught 
that ivell, before he is able to compare different 
methods with each other. 



ORDER OF THE SUBJECTS ARITHMETIC. 

§ 374. Arithmetic is the most useful and Arithmetic: 
simple branch of mathematical science, and is 
the first to be taught. If, however, the pupil 
has time for a full course, I would by no means connection 
recommend him to finish his Arithmetic before Algebra. 
studying a portion of Algebra. 



346 



APPENDIX. 



ALGEBRA. 

Algebra: § 3 ?5. Algebra is but a universal Arithmetic, 
with a more comprehensive notation. Its ele- 
ments are acquired more readily than the higher 
and hidden properties of numbers ; and indeed 
the elements of any branch of mathematics are 
more simple than the higher principles of the 
now preceding subject ; so that all the subjects can 

il should bo 

studied: best be studied in connection with those which 
preceae and follow* 

should § 37G. Algebra, in a regular course of instruc- 

preoede 

Geometry: tion, should precede Geometry, because the ele- 
mentary processes do not require, in so high a 
Why. degree, the exercise of the faculties of abstrac- 
tion and generalization. But when we have 
when completed the equation of the second degree, 
should be the processes become more difficult, the abstrac- 
commenced. t j ons more perfect, and the generalizations more 
extended. Here then I would pause and com- 
mence Geometry. 

GEOMETRY. 

Geometry. § 377. Geometry, as one of the subjects of 
mathematical science, has been fully considered 
in Book II. It is referred to here merely to mark- 
its place in a regular course of instruction. 



APPENDIX. 347 



TRIGONOMETRY — PLANE AND SPHERICAL. 

§ 378. The next subject in order, after Geom- Trigonome- 
try: 

etry, is Trigonometry : a mere application of the 
principles of Arithmetic, Algebra, and Geometry what u is. 
to the determination of the sides and angles of 
triangles. As triangles are of two kinds, viz. 
those formed by straight lines and those formed 
by the arcs of great circles on the surface of a 
sphere ; so Trigonometry is divided into two Two kii*u. 
parts : Plane and Spherical. Plane Trigonom- 
etry explains the methods, and lays down the P lane - 
necessary rules for finding the remaining sides 
and angles of a plane triangle, when a sufficient 
number are known or given. Spherical Trigo- spherical, 
nometry explains like processes, and lays down 
similar rules for spherical triangles. 



SURVEYING AND LEVELLING. 

§ 379. The application of the principles of 
Trigonometry to the measurement of portions 
of the earth's surface, is called Surveying ; and surveying, 
similar applications of the same principles to the 
determination of the difference between the dis- 
tances of any two points from the centre of the 
earth, is called Levelling. These subjects, which LeTeMng. 
follow Trigonometry, not onlv embrace the va- 



348 APPENDIX. 



What they rious methods of calculation, but also a descrip- 
tion of the necessary Instruments and Tables. 
They should be studied immediately after Trigo- 
nometry ; of which, indeed, they are but appli- 
cations. 

DESCRIPTIVE G E O M £ T R V . 

Descriptive § 380. Descriptive Geometry is that branch 

Geometry : 

of mathematics which considers the positions of 
the geometrical magnitudes, as they may exist in 

space, and determines these positions by refer- 
ring the magnitudes to two planes called the 

Planes of Project inn. 

itsnature. it is, indeed, hut a development of tho 

eral methods, by which lines, surfaces, and solids 
may be presented to the mind by means of 
drawings made upon paper. The processes of 

what its this development require the constant exercise of 
the conceptive faculty. All geometrical mag- 
nitudes may be referred to two planes of pro- 
jection, and their representations on these planes 
will express to the mind, their forms, extent, and 
also their positions or places in space. From 
How. these representations, the mind perceives, as it 
were, at a single view, the magnitudes them- 
selves, as they exist in space ; traces their boun- 
daries, measures their extent, and sees all their 
parts separately and in their connection. 



■tudy aivom 
plishes. 



APPENDIX. 349 



In France, Descriptive Geometry is an impor- How 
tant element of education. It is taught in most France . 
of the public schools, and is regarded as indis- 
pensable to the architect and engineer. It is, 
indeed, the only means of so reducing to paper, 
and presenting at a single view, all the compli- 
cated parts of a structure, that the drawing or 
representation of it can be read at a glance, and 
all the parts be at once referred to their appropri- 
ate places. It is to the engineer or architect not its value 
only a general language by which he can record branch. 
and express to others all his conceptions, but is 
also the most powerful means of extending those 
conceptions, and subjecting them to the laws of 
exact science. 



SHADES, SHADOWS, AND PERSPECTIVE. 

§ 381. The application of Descriptive Geom- 
etry to the determination of shades and shadows, shades, 
as they are found to exist on the surfaces of ^ 
bodies, is one of the most striking and useful ap- Pers P ectiY<v 
plications of science ; and when it is further 
extended to the subject of Perspective, we have 
all that is necessary to the exact representation 
of objects as they appear in nature. An accu- 
rate perspective and the right distribution of 
light and shade are the basis of every work of 



350 APPENDIX. 



Their use. the fine arts. Without them, the sculptor and 
the painter would labor in vain : the chisel of 
Canova would give no life to the marble, nor the 
touches of Raphael to the canvas. 



ANALYTICAL GEOMETRY. 

Analytical § 382. Analytical Geometry is the next sub- 
Geometry. j ec j. -^ a re g U ] ar course f mathematical study, 

though it may be studied before Descriptive Ge- 
ometry. The importance of this subject cannot 
its be exaggerated. In Algebra, the symbols of 

importance: . 11 i 

quantity have generally so close a connection 
with numbers, that the mind scarcely realizes 
Valuable as the extent of the generalization; and the power 
a8Uy ' of analysis, arising from the changes that may 
take place among the quantities which the sym- 
bols represent, cannot be fully explained and de- 
veloped. 

But in Analytical Geometry, where all the 
Ret* us. magnitudes are brought under the power of anal- 
ysis, and all their properties developed by the 
combined processes of Algebra and Geometry, we 
are brought to feel the extent and potency of 
those methods which combine in a single equa- 
tion every discovered and undiscovered property 
of every line, straight or curved, which can be 
formed by the intersection of a cone and plane. 



APPENDIX. 351 



To develop every property of the Conic Sec- its extent 
tions from a single equation, and that an equa- 
tion only of the second degree, by the known 
processes of Algebra, and thus interpret the re- 
sults, is a far different exercise of the mind from 
that which arises from searching them out by 
the tedious and disconnected methods of separate 
propositions. The first traces all from an inex- lis methods 
haustible fountain by the known laws of analyti- 
cal investigation, applicable to all* similar cases, 
while the latter adopts particular processes ap- 
plicable to special cases only, without any gen- 
eral law of connection. 



DIFFERENTIAL AND INTEGRAL CALCULUS. 

§ 383. The Differential and Integral Calculus Differential 
presents a new view of the power, extent, and j^g^ 
applications of mathematical science. It should Calculus - 

be carefully studied by all who seek to make what per- 

... . . . 'ii it s°ns should 

high attainments in mathematical knowledge, or gt udyit. 
who desire to read the best works on Natural 
and Experimental Philosophy. It is that field of 
mathematical investigation, where genius may 
exert its highest powers and find its most certain 
rewards. 



INDEX. 



Abstraction That faculty of the mind which enables us, in contem- 
plating any object to attend exclusively to some par- 
ticular circumstance, and quite withhold our attention 
from the rest, Section 12. 
" Is used in three senses, 13. 

Abstract Quantity, 75, 96. 

Addition, Readings in, 116. 

■ Examples in, 151. 

" of Fractions, Rule for, 191. 

" Combinations in, 192, 193. 

Definitions of, 203. 
" One principle governs all operations in, 232. 

JEtna, How far designated by the term mountain, 20. 

A Geometrical Proportion, 168. 

Algebra A species of Universal Arithmetic, in which letters and 

signs are employed to abridge and generalize all pro- 
cesses involving numbers, 280. 
" Divided into two parts, 2 SO. 

■ Difficulties of, from what arising, 286. 

" Principles of, deduced from definitions and axioms, 297. 

■ Should precede Geometry in instruction, 376. 
Alphabet of the language of numbers, 80, 113, 114-. 

" Language of Arithmetic, formed from, 192. 

Analytical Form, for what best suited, 71, 89. 

Analysis . . . A term embracing all the operations that can be performed 

on quantities represented by letters, 87, 88, 274, 327. 

* It also denotes the process of separating a complex whole 

into its parts, 89. 

* of problems in Arithmetic, 175, 176. 

23 



354 



INDEX. 



Analysis Three branches of, Sections 279, 285, 286. 

" First notions of, how acquired, 317. 

u Problems it. has solved, 344-347. 

Angles Right angle, the unit of, 250. 

" A class of Geometrical Magnitudes, 273. 

Apothecaries' Weight — Its units and scale, 138. 

Apprehension Simple apprehension is the notion (or conception) of an 

object in the mind, 7. 

" Incomplex apprehension is of one object or of •erend 

without any relation being perceived between them, 7. 

" Complex is of several with such a relation, 7. 

Area or CONTENTS, Number of times a surface contains its unit* of 

measure, Ml. 
Argument With one premiss suppressed ifl called an Knthvmeme, 47. 

" Two kinds of objections to an, 17. 

" Every valid, may l>e reduced to a syllogism, 52. 

" at full length, a syllogism, 56. 

" concerned with connection between premises and conclu- 

sion, 57. 

" Where the fault (if any) fies, ♦>'.>. 

Arguments, In reasoning we make UBS of, 1_. 

" Examples of unsound, 50. 

" Rules for examining, 70. 

Aristotle did not mean that arguments should always be stated 

syllogistically, 53. 

* accused of darkening his demonstrations by the use of 

symbols, 57 
u His philosophy not progressive, SS4, 

Aki3totles Dictum — Whatever is predicated (that is, affirmed or de- 

nied) universally, of any class of things, may 
be predicated, in like manner (viz. affirmed 
or denied), of any thing comprehended in 
that class, 54. 

• " Keystone of his logical system, 54. 
" " Objections to, 54, 55. 

" " a generalized statement of all demonstration, 55. 

■ u applied to terms represented by letters, 56. 

«* " not complied with, 59, 60. 

" " All sound arguments can be reduced to the form 

to which it applies, 65, 66. 



INDEX. 355 



Arithmetic Is both a science and an art, Section 172. 

" It is a science in all that relates to the properties, laws, 

and proportions of numbers, 172. 
" It is an art in all that concerns their application, 173. 

" Processes of, not affected by the nature of the objects, 43. 

" Illustration from, 45. 

" How its principles should be explained, 174 

" Its requisitions as an art, 177. 

* Faculties cultivated by it, 180. 
a Application of principles, 188. 

■ Generally preceded by a smaller treatise, 190 
u Methods of placing subjects examined, 191. 

" Combinations in, 192-199. 

■ What its study should accomplish, 206. 
" Art of, its importance, 206. 

" Elementary ideas of, learned by sensible objects, 207 

a Principles of, how they should be taught, 208. 

u First, what it should accomplish, 214. 

* " arrangement of lessons, 214-223. 
" " what should be taught in it, 226. 

" Second, should be complete and practical, 227. 

" u arrangement of subjects, 228. 

* " introduction of subjects, 229. 

u " reading of figures should be constantly prac- 

tised, 230. 
u Third, the subject now taught as a science, 231. 

u " requirements from the pupil for, 231. 

" " Reduction and the ground rules brought under one 

principle, 232. 
u " design of, — methods must differ from smaller 

works, 233. 
u " examples in the ground rules, 234. 

" " what subjects should be transferred from elemen- 

tary works, 235. 
" Practical utility of, 356, 357. 

" should not be finished before Algebra is commenced, 374. 

Arithmetical Proportion, 163. 

Ratio, 163. 

Art The application of knowledge to practice, 22. 

¥ Its relations to science, 22. 



356 



IN DEX. 



Art. 

u 

Astronomy 

u 
u 

Authors, 



Auxiliary 
Avoirdupois 

Axiom. ... 

Axioms 



.A single one often formed from several sciences, Section 22, 
of Arithmetic, 173, 177, 182. 

brought by Newton within the laws of mechanics, 337. 
How it became deductive, 339. 
Mathematics necessary in, 341. 
methods of finding ratio, 165, 170. 

" of placing Rule of Three, 187. 
quotations from, on Arithmetic, 201-20-4 
definition of proportion, 268. 
Quantities, 259, 261. 
Weight, its units and scale, 136 
.A self-evident truth, 27, 97. 
of Geometry, process of learning them, 27. 
or canons, for testing the validity of syllogisms, 67. 
of Geometry established by Induction, 73. 
for forming numbers, 79. 

for comparison relate to equality and inequality, 102. 
for inferring equality, 102, 258, 260, 264. 

" inequality, 102. 
employed in solving equations, 278, 311. 



Bacon, Lord, 



Barometer, 
Barrow, Dr., 
Belief 

Blakewell, 
Bowditch, 

Breadth 

Bridge, 



Quotation from, 328. 

Foundation of his Philosophy, 334 ; its subject Nature 

335, page 12. 
His system inductive, 334. 
Object and means of his philosophy, 335 
Construction and use of, 359 
Quotation from, 328, 340. 
essential to knowledge, 23. 
and disbelief are expressed in propositions, 36. 
steps of his discovery, 32. 
Tables of, used in Navigation, 359. 
.A dimension of space, 82. 
Harlem, description of, 362. 



Calculus, 



Canons 
Cause 



.In its general sense, means any operation performed on 

algebraic quantities, 281, 282. 
Differential and Integral, 283-285, 383. 
for testing the validity of syllogisms, 67. 
and effect, their relation the scientific basis of induction, 33. 



INDEX. 



3o7 



Chemist, 
Chemistry 
Chicle 



Circular Measure. 
Classes 



Classification. 



Coefficient 

Coins 
Combinations 

Comets, 
Comparison, 



Conclusion. . 



Concrete 
Conjunctions 

Constants . . 



Copula 



Cousin, 
Curves, 
Croton 



Illustration, Section 53 ; idea of iron, 322. 
aided by Mathematics, 342. 
.A portion of a plane included within a curve, all the 

points of which are equally distant from a certain point 

within called the centre, 244. 
The only curve of Elementary Geometry, 244. 
Property of, 256. 
its units and scale, 149. 

.Divisions of species or subspecies, in which the charac- 
teristic is less extensive, but more full and complete, 16. 
.The arrangement of objects into classes, with reference to 

some common and distinguishing characteristic, 16. 
Basis of, may be chosen arbitrarily, 20. 
of a letter, 291 ; of a product, 292. 
Differential, 283, 284. 

should be exhibited to give ideas of numbers, 133. 
in Arithmetic, 192-199. 
taught in First Arithmetic, 216-218. 
Problem with reference to, 347. 
Knowledge gained by, 95. 
Reasoning carried on by, 25, 307. 
.The third proposition of a syllogism, 40. 
in Induction, broader than the premises, 31. 
deduced from the premises, 40, 41, 46, 47, 49. 
contradicts a known truth, in negative demonstrations, 

264, 265. 
Quantity, 75, 96. 
causal, illative, 48. 

denote cause and effect, premiss and conclusion, 48. 
.Quantities which preserve a fixed value throughout the 

same discussion or investigation, 282, 283, 313. 
represented by the first letters of the alphabet, 284 
.That part of a proposition which indicates the act of 

judgment, 38. 
must be "is" or "is not," 38, 39. 
quotation from, 180. 

circumference of circle the simplest of, 239. 
river, its sources, 362. 

dam, its construction, 362 ; lake, area of, 362 
aqueduct, description of, 362. 



358 



INDEX. 



Decimals, 
Deduction 



Deductive 

M 

Definition 

M 

4« 

Definitions, 



Demonstration 



Descartes, 

Dictum, 

Differential 



Discussion 
Distribution 



language and scale for, Sections 156, 157. 
,.A process of reasoning by which a particular truth is in- 
ferred from other truths which are known or admitted, 34, 

Its formula the syllogism, 34. 

Sciences, why they exist, 98. 

" aid they give in Induction, 335. 

.A metaphorical word, which literally signifies laying 
down a boundary, 1. 

Is of two kinds, 1. 

Its various attributes, 2-5. 

General method of framing, 3. 

Rules for framing, 5 (Note). 

and axioms, tests of truth, 97, 99. 

signs of elementary ideas, 200. 

Necessity of exact, 200. 

.A series of logical arguments brought to a conclusion, in 
which the major premises are definitions, axiom-, or 
propositions already established, 237. 

of a demonstration, 55. 

to what applicable! 238. 

of Proposition I. of Legendre, 258. 

positive and negative, 2G2-2G5. 

produces the most certain knowledge, 326. 

originator of Analytical Geometry, 281 

Aristotle's, 54, 55, G6. 

and Integral Calculus. The science which notes the 
changes that take place according to fixed laws estab- 
lished by algebraic formulas, when those changes are 
indicated by certain marks drawn from the variable 
symbols, 283. 

Coefficients — Marks drawn from the variable symbols, 
283, 284. 

and Integral Calculus — Difference between it and Ana- 
lytical Geometry, 284. 
" " " What persons should study it, 383, 

of an Equation, 308. 

. .A term is distributed, when it stands for all its signift 
cates, 61. 

A term is not distributed when it str.nds for only a part 
of its significates, 61. 



INDEX. 



359 



Distribution, 
Division, 



Dry Measure, 
Duodecimal 



Words which mark, not always expressed, Section 62. 

Readings in, 123 ; examples in, 154. 

Combinations in, 196, 

All operations in, governed by one principle, 232. 

of quantities, how indicated, 294. 

Its units and scale, 147. 

units, 142-144. 



English Money, Its units and scale, 135. 

Enthymeme An argument with one premiss suppressed, 47. 

Equal. Two geometrical figures are said to be equal when they 

can be so applied to each other as to coincide through- 
out their whole extent, 255, 312. 

Equality In Geometry expresses that two figures coincide. In 

Algebra it merely implies that each member of an 
equation contains the same unit an equal number of 
times, 312. 

Equation An analytical formula for expressing equality, 307-3 1 2. 

" A proposition expressed algebraically, in which equality 

is predicated of one quantity as compared with an 
other, 309. 
" either abstract or concrete, 310. 

Equations, subject of, divided into two parts, 308. 

" Five axioms for solving, 311. 

Equivalent Two geometrical figures are said to be equivalent when 

they contain the same unit of measure an equal num- 
ber of times, 255. 

Examples in ground rules of Third Arithmetic, 234. 

Of little use to vary forms of, without changing the prin- 
ciples of construction, 236. 

Experiment, in what sense used, 25 (Xote). 

Exponent An expression to show how many equal factors are «ia 

ployed, 293. 

Extremes. Subject and predicate of a proposition, 38, 67. 

Fact Any thing which has been or is, 24. 

" Knowledge of, how derived, 25. 

" In what sense used, 25. 

" regarded as a genus, 25. 

Factories, value of science in, 358. 



360 



INDEX 



Fallacy Any unsound mode of arguing which appears to demand 

our conviction, and to be decisive of the question ic 
hand, when in fairness it is not, Section 68. 
Illustration of, 53. 

■ Example and analysis of, 50, 50. 
" Material and Logical, 69. 

" Rules for detecting, ^70. 

Federal Money, units increase by scale of tens, 129, 134. 

■ Methods of reading, 129, 184 

Figure A portion of space limited by boundaries, 83. 

" Each geometrical, stands for a class, 277. 

Figures in Arithmetic show how many times a unit is taken, 125. 

" do not indicate the kind of unit, 125. 

Laws of the places of, 126, 15 

" have no value, 198, 201. 

" Methods of reading, 130; of writing, 199. 

Definitions of, 201, 202. 

" should be early used in Arithmetic, 219. 

First Arithmetic, what should he taught in it, 226. 

" Faculties to be cultivated DJ it, 211. 

■ Construction of the learoni, 211-218. 
" Lesson in Fraction*, 220 -224. 

" Tables of Denominate Numbers — Example*, 225. 

Fractions come from the unit one, 182. 

■ should be constantly compared with one, 162. 

* Reasons for placing Common Fractions immediately aftcf 

Division examined, 1S9. 

■ not " unexecuted divisions," 189. 
u Elementary idea of, 189. 

■ Expression for, the same as for Division, 189. 
« Definitions of, 204. 

• Lessons in, in First Arithmetic, 220-224. 
Fractional units, 155 ; orders of, 156 ; language of, 15G-159, 197. 

" ■ three things necessary to their apprehension, 160. 

" " advantages of, 161. 

" " two things necessary to their being equal, 161. 



Galileo, imprisoned in the 17th century, 343. 

Generalization.... The process of contemplating the agreement of several 
objects in certain points, and giving to all and caeli Ofl 



INDEX. 



361 



these objects a name applicable to them in resp^ 
this agreement, Be : 
Generalization implies abstraction, 14. 

" must be preceded by knowledge, I 

Gent s The most extensive term of classification, and conse- 
quently the one involving the fewest particulars, I 
" Highest. That which cannot be referred to a more ex- 

tended classincaticr. 
■ Subaltern*. A :f a more extended classifica- 

tion, I & 
Geometrical Magnitudes, three classes of. 2 

* "do not involve matter, '247. 

" ■ their boundaries or limits. - r" 

u u each has its unit of measure, . 

■ ** analysis of comparison, 270. 271. 

* "to what the examination of properti-- 

reference. 

■ Proportion, 163 ; Ratio, 163 ; Progression, 170. 
Geo^itry Treats of space, and compares portions of space with each 

other, for the purpose of poi: :heir prop 

and mutual relati*. - 

■ Why a deductive science. . 

■ : notions of. how acquired. 
u Practical utility of. 

a Origin of the E 

u Its place in a course of instruction. 

* Analytical. RwBWBffl the properties, measures, and re 

lations of the Geometrical Magnitudes by 
means of the analytical symbols 1 1 
u u originated w; 281. 

■ ■ difference between it and Calculus. - 

M ■ Hb importan : end metho . « . . ? _ 

* ■ That branch of mathematics which eon- 

Magniru . f exist in space. 

and deternik positions by re- 

ferring the magnitudes to two planes 
called the Planes of Projection. I - . 

■ * how regarded in France. 
Governor. 



362 



INDEX 



Grammar 
Gravitation, 



defined, Section 113. 
Law of, 32, 344. 



Hall, Captain's, voyage from San Bias to Rio Janeiro, 359. * 

Harlem river, Bridge over, and width, 362. 

Herschel, Sir John, Quotation from, 27, 322, 341, 359. 
Hull of the steamship, how formed, 359. 

Illative Conjunctions, 48. 

Illicit Process When a term is distributed in the conclusion which waa 

not distributed in one of the premises, 67. 
Indefinite Propositions, 02. 

Index of a root, 205. 

Induction Is that part of Logic which infers truths from facts, 30-33. 

Logic of, 30. 
" supposes necessary observations accurately made, 32. 

Example of, Blakewell, 82 ; of Newton, 32. 
44 based upon the relation of cause and effect, 33. 

44 Reasoning from particulars to generals, 34. 

44 its place in Logic, 72. 

44 how thrown into the form of a syllogism, 74, 99. 

" Truths of, verified by Deduction, Co5, 336. 

Inertia proportioned to weight, 26S. 

Infinity, The limit of an increasing quantity, 002-306. 

Integer Numbers, why easier than fractions, 102. 

" constructed on a single principle, 231. 

Intuition Is strictly applicable only to that mode of contemplation, 

in which we look at facts, or classes of facts, and im- 
mediately apprehend their relations, 27. 
Iron, different ideas attached to the word, 322. 

Judgment Is the comparing together in the mind two of the notions 

(or ideas) which are the objects of apprehension, and 
pronouncing that they agree or disagree, 8. 
44 is either Affirmative or Negative, 8. 



Kant, quotation from, 21. 

Knowledge Is a clear and certain conception of that which is true, 23. 

44 facts and truths elements of, 25. 

of facts, how derived, 25. 



INDEX. 



363 



Knowledge some possessed antecedently to reasoning, Section 29. 

" the greater part matter of inference, 29. 

" must precede generalization, 184. 

" two ways of increasing, 323. 

" cannot exceed our ideas, 323. 

" the increase of, renders classification necessary, page 20. 



Language Affords the signs by which the operations of the mind are 

recorded, expressed, and communicated, 10. 
" Every branch of knowledge has its own, 11. 

" of numbers, 80; of mathematics, 91, 92. 

* of mathematics must be thoroughly learned, 92. 

" " " its generality, 93. 

" for fractional units, 156, 159, 197. 

Arithmetical, 192-199. 
" exact, necessary to accurate thought, 205. 

** of Arithmetic, its uses, 219. 

" of Algebra, the first thing to which the pupil's mind 

should be directed, 290. 
" Culture of the mind by the use of exact, 322. 

Science makes them known, 21, 315. 

" refers individual cases to them, 55. 
generalized facts, 55, page 14. 
include all contingencies, 332. 
every diversity the effect of, 34G. 
one dimension of space, 81. 
in First Arithmetic, how arranged, 214. 
" " " their connections, 218. 

may stand for all numbers, 276. 
" represents things in general, 277. 

Levelling The application of the principles of Trigonometry to the 

determination of the difference between the distances 
of any two points from the centre of the earth, 379. 
" Its practical uses, 360. 

Limit, definition of, 306. 

Line One dimension of space, 83, 239. 

■ A straight line does not change its direction, 83, 239, 318. 

■ Curved line, one which changes its direction at every 

point, 83, 239. 
u Axiom of the straight, 239. 



Laws of Nature. 



Length 
Lessons 



Letter 



364 INDEX. 



Lines, limits of, Section 247 

" Auxiliary, 259. 

Liquid Measure, Its units and scale, 146. 
" Local value of a figure," has no significance, 128, 201. 
Locke, Quotation from, 323. 

Logic Takes note of and decides upon the sufficiency of the evi- 
dence by which truths are established, 29. 

" Nearly the whole of science and conduct amenable to, 29 

" of Induction, its nature, 30. 

" Archbishop AYhateley's views of, 72. 

Mr. Mill's views of, 72. 
Logical Fallacy, 69. 

Machinery of factories Arranged on a general plan, 858. 

" of the steamship) £5& 

Major Premiss, often suppressed, cannot be denied, 46. 

" ultimate, of Induction, 71, 99. 

Major Premises of Geometry, 287, 267. 
Mansfield, Mr., Quotation from, 326, Wfc 

Mark The evidence contained in the attributes implied in a 

general name, by which we infer that any lllillg called 
by that name ; another attribute or set of at- 

tributes. For example : " All equilateral triangles are 
equiangular." Knowing this general proposition, when 
we consider any object possessing the attributes implied 
in the term "equilateral triangle," we may infer that it 
possesses the attributes implied in the term " equian- 
gular;" thus using the first attributes as a mark or 
evidence of the second. Hence, whatever pos-. 
any mark possesses those attributes of which it h a 
mark, 98, 257 250. 
Masts of the steamship, how placed, 050. 

Material Fallacy, 69. 
Mathematical "Reasoning conforms to logical rules, 73. 

" " ever} r truth established by, is developed by a 

process of Arithmetic, Geometry, or Analy 
sis, or a combination of thein 90. 

Mathematics The science of quantity, 76. 

u Pure, embraces the principles of the science, 76-78 

'* " on what based, 97. 



INDEX. 365 



Mathematics Mixed, embraces the applications, Section 76. 

" Primary signification, 77. 

u Language of, 91. 

u " Exact science," 97. 

** Logical test of truth in, 97. 

" a deductive science, 97, 98. 

u concerned with number and space, 73, 76, 78, 101. 

" What gives rise to its existence, 100. 

" Why peculiarly adapted to give clear ideas, 324-326, 329 

** a pure science, 329. 

* considered as furnishing the keys of knowledge, 331. 
u Widest applications are in nature, 334. 

M Effects on the mind and character, 328, 340. 

u Guidance through Xature, 340. 

" Its necessity in Astronomy, 341. 

« Results reached by it, 349, 350. 

* Practical advantages of, 355. 

" What a course of, should present, and how, 365, 366. 

■ Reasonings of, the same in each branch, 367. 

" Faculties required by, 369. 

" Necessity of, to the philosopher, page 16. 

Measure A term of comparison, 94. 

" Unit of, should be exhibited to give ideas of numbers, 133 
" " for lines, surfaces, solids, 249. 

u of a magnitude, how ascertained, 249. 

Middle Term distributed when the predicate of a negative proposi- 
tion, 64. 

" When equivocal, 67. 

Mill, Mr. his views of Logic, 72, 74. 

Mind, Operations of, in reasoning, 6. 

" Abstraction a faculty, process, and state of, 13. 

K Processes of, which leave no trace, 68. 

■ Faculties of, cultivated by Arithmetic, 180. 

■ Thinking faculty of, peculiarly cultivated by mathemat* 

ics, 325, 326. 

Minus sign, Power of, fixed by definition, 297. 

Motion proportional to force impressed, 26S. 
Multiplication, Readings in, 122 ; examples in, 153. 

■ What the definition of, requires, 177. 

" Combinations in, 195. 



366 INDEX. 



Multiplication, All operations in, governed by one principle, Section 232. 
" in Algebra, illustrations of, 299-301. 

Names, Definitions are of, 1. 

" given to portions of space, and defined in Geometry, 238. 

Naturalist determines the species of an animal from examining a 

bone, 333. 
Negative premises, nothing can be inferred from, 67. 

" demonstration, its nature, 263, 2G5 ; illustration of, 264. 

Newton, his method of discovery, 32. 

" changed Astronomy from an experimental to a deductive 

science, 337, 339. 
Non-distribution of terms, 61. 

" Word "some" which marks, not always expressed, 62 

Numbers Are expressions for one or more tlungs of the same kind, 

79, 106. 
" How learned, 79. 

" Axioms for funning, 79, 304. 

• Three ways of expressing, 107. 

■ Ideas of, complex, 108, 12-1. 

u Two things necessary for apprehending clear] y, 110. 

■ Simple and Denominate, 112. 

u Examples of reading Simple, 130. 

■ Two ways of forming from one, 131. 

** first learned through the senses, 133, 316. 

** Two ways of comparing, 163. 

u compared, must be of the same kind, 171, 175. 

" Definitions of, 201, 202. 

" must be of something, 275. 

" may stand for all things, 27o. 

" First lessons in, impress the first elements of mathemati- 
cal science, 370. 

Olmsted's Mechanics, quotation from, 269. 

Optician, Illustration, 212. 

Oral Arithmetic, its inefficiency without figures, 219. 

Order of subjects in Arithmetic, 182, 188. 

Parallelogram ...A quadrilateral having its opposite sides taken two and 
two parallel, 242. 



INDEX. 



367 



Parallelogram regarded as a species, Section 17 as a genus, 18. 

" Properties of, 256. 

Particular proposition, 62. 

■ premises, nothing can be proved from, 67. 
Pendulum, the standard for measurement, 253. 
Philosophy, Natural, originally experimental, 337. 

■ " has been rendered mathematical, 337. 
Place idea attached to the word, 81. 

u designates the unit of a number, 202. 

Plane That with which a straight line, having two points m 

common, and any how placed, will coincide, 240. 
" First idea of, how impressed, 319. 

Plane Figure ... .Any portion of a plane bounded by lines, 2-10. 
Plane Figures in general, 243. 

Point That which has position in space without occupying any 

part of it, 81. 
Points, extremities or limits of a line, 239. 

Practical Rules in Arithmetic, 177, 178. 

" The true, 207, must be the consequent of science, 228. 

* Popular meaning of, 351, 353. 

" Questions with regard to, 351, 352. 

" Consequences of an erroneous view of, 354. 

■ True signification of, 354. 
precedes theory, but is improved by it, 42, 
without science is empiricism, page 13. 

...That which is affirmed or denied of the subject, 38 
" Distribution, 63. 

" Xon -distribution, 63. 

■ sometimes coincides with the subject, 63. 

Premiss Each of two propositions of a syllogism admitted to be 

true, 40. 
Major Premiss — The proposition of a syllogism which 

contains the predicate of the conclusion, 40. 
Minor Premiss — The proposition of a syllogism which 
contains the subject of the conclusion, 40. 
5sure, a law of fluids, 364. 

,ciple of science applied, 22 

■ on which valid arguments are constructed, 52 
" Value of a, greater as it is more simple, 54. 

• Aristotle's Dictum, a general, 55. 



Practice 



Predicate . 



368 



INDEX. 



Principle 



Principles 



Process 
Product 
Progression, 
Property 
Proportion . 



Proposition 



the same in the ground rules for simple and denominate 
numbers, Sections 151-154, 232. 

of science and rule of art, 179. 

should be separated from applications, 186, 187. 

of science are general truths, 208. 

of Arithmetic, how taught, 208. 

should precede practice, 229. 

of Mathematics, deduced from definitions and axioms, 297 

of acquiring mathematical knowledge, 316-320. 

of several numbers, 292. 

Geometrical, 170. 

of a figure, 256. 

.The relation which one quantity bears to another with re- 
spect to its being greater or less, 103, 267-269 

Arithmetical and Geometrical, 163. 

Reciprocal or Inverse, 269. 

of geometrical figures, -7' 
..A judgment expressed in words, 85. 

All truth and all error lie in propositions, also answers to 
all questions, :)0. 

formed by patting together two names, 37. 

CMiiM^ts of three par: - 

subject, and predicate, called extremes, 38. 

Affirmative, 39 ; Negative, 

Three proposition- essential to a syllogism, 10. 
Universal, 02. 

Particular, 62. 



Quadrilateral A portion of a plane bounded by four straight lines, 242. 

" regarded as a genus, 17. 

" Different varieties of, 212. 

Quality of a proposition refers to its being affirmative or nega- 

tive, 63. 
Quantities only of the same kind can be compared, 267. 

" Two classes of, in Algebra, 287, 313. 

" " " " in the other branches of Analysis, 282, 

2S3, 313. 
u compared, must be equal or unequal, 102, 307. 

Quantity Is a general term applicable to everything which can 

be increased or diminished, aid measured, 75. 82k 



INDEX. 



369 



Quantity, Abstract, does not involve matter, Sections 75, 96 

" Concrete does, 75, 96, 

" Propositions divided according to, 62. 

" presented by symbols, 93. 

" consists of parts which can be numbered, 276 

w Constant, 282. 

" Variable, 282. 

" Five operations can be performed on, 288, 295. 

" represented by five signs, 289. 

" Nature o£ not affected by the sign, 290, 296. 

Questions known, when all propositions are known, 36. 

■ with regard to number and space, 78. 

Analysis of, 175, 176. 
•' Difficult, in Fractions avoided, 191 

" with regard to methods of instruction, 371. 

Quotations from Kant, 21 ; Sir John Herschel, 27, 322, 341, 359 ; 

Cousin, 180; Olmsted's Mechanics, 268; Locke, 323; 
Mansfield's Discourse on Mathematics, 325, 327 ; Lord 
Bacon, 328 ; Dr. Barrow, 328, 340. 



Railways, Problem presented in, 361. 

Rainbow, Illustration, 322. 

Ratio The quotient arising from dividing one number or quan- 
tity by another, 163, 267. 
" Discussion concerning it, 165-171. 

" Arithmetical and Geometrical, 163. 

■ How determined, 165. 

" An abstract number, 267, 272. 

" Terms direct, inverse, or reciprocal, not applicable to, 269. 

Reading in Addition, 116, 117 ; advantages of, 118. 

" in Subtraction, 120. 

" in Multiplication, 122, 

" in Division, 123. 

* of figures, its aid in practical operations, 230. 

Reason, To make use of arguments, 42. 

" A premiss placed after the conclusion, 48. 

Reasoning; The act of proceeding from certain judgments to another, 

founded on them, 9. 
" Three operations of the mind concerned in, 6. 

Process, sameness of the, 42, 43, 45, 314. 

24 



370 INDEX. 



Reasoning processes of mathematics consist of two parts, Section 78. 

" in Analysis is based on the supposition that we are deal- 

ing with things, 278. 
Reciprocal or Inverse Proportion, 269. 

Rectangle A parallelogram whose angles are right angles, 242. 

Remarks, Concluding subject of Arithmetic, 236. 

Reservoirs, Croton, description of, 362. 
Right angle Definition of, 258. 

Roman Table, when taught, 215. 
Root, Symbol for the extraction of, 205 

Rule of Three, Solution of questions in, 169. 
" Comparison of numbers, 186. 

" should precede its applications, 187. 

Rules, Every thing done according to, 21. 

" of reasoning analogous to those of Arithmetic, 45 

u Advantages of logical, 50. 

" for teaching, 186. 

How framed, 297. 

Scale of Tens, Units increasing by, 124-130, 157, 1S3. 

Science In its popular sense means knowledge reduced to order 

21, 326. 

" In its technical Bense means an analysis of the laws of 

nature, 21. 

" contrasted with art, 22. 

" of Arithmetic, 172. 

" Principles of, 200, 208. 

** Methods of, must be followed in Arithmetic, 228. 

" of Geometry, 237, 248, 257. 

■ Objects and means of pure, 322. 

u should be made as much deductive as possible, 336. 

u Deductive and experimental, 337. 

u when experimental, 33S, 339 ; when deductive, 338, 339. 

u What it has accomplished, 348. 

" Practical value of, in factories, 358. 

■ " " " in constructing steamships, 359. 

« " " " in laying out and measuring land, 360. 

" " " " in constructing railways, 361. 

* Its power illustrated in Croton aqueduct, 362. 

• What constitutes it, 372. 






INDEX. 371 



Second Arithmetic, its place and construction, Section 227-230. 
Sextant, its uses in Navigation, 359. 

Shades, Shadows, and Perspective — An application of Descriptive Geom- 
etry, 381. 

Significate An individual for which a common term stands, 15. 

Signs, Five used to denote operations on quantity, 289. 

" How to be interpreted, 290. 

" do not affect the nature of the quantity, 290, 296. 

" indicate operations, 296, 298. 

Solid A portion of space having three dimensions, 85. 

" A portion of space combining the three dimensions of 

length, breadth, and thickness, 246, 320 
« Limit of, 2±1. 

" First idea of, how impressed, 320. 

Solids bounded by plane and curved surfaces, 85. 

" Three classes of, 246. 

" Analysis of comparison, 271, 272. 

" Comparison of, under the supposition of changes in their 

volumes, 272. 
Solution of all questions in the Rule of Three, 169. 

" of an equation in Algebra, 308. 

Space Is indefinite extension, 81, 82. 

" has three dimensions, length, breadth, and thickness, 82 

" Clear conception of, necessary to understand Geometry, 

238. 
Species One of the divisions of a genus in which the characteris- 
tic is less extensive, but more full and complete, 16, 17. 
Subspecies — One of the divisions of a species, in which 
the characteristic is less extensive, but more full and 
complete, 16, 19. 
Lowest Species — A species winch cannot be regarded 
as a genus, 17. 
Spelling, 113 ; in Addition, <fcc., 115-123. 

Square A quadrilateral whose sides are equal, and angles right 

angles, 242. 
Statement of a proposition in Algebra, 308, 

■ in what it consists, 309. 

Steamship, an application of science, 359. 

Subject The name denoting the person or thing of which some- 
thing is affirmed or denied, 38. 



372 INDEX. 



Subjects, How presented in a text-book, Section 209-212. 

Subtraction, Readings in, 120. 

" Examples in, 152. 

" Combinations in, 194. 

" All operations in, governed by one principle, 232 

" in Algebra, illustration of, 298. 

Suggestions for teaching Geometry, 273. 

" for teaching Algebra, ol5. 

Sum, Its definition, 203. 

Surface A portion of space having two dimensions, 84, 240, 319 

Plane and Curved, 84, 2 10. 
Surfaces, Curved, 245. 

" " of Elementary Geometry, 245. 

LimiU of, 247. 

Sirvkyinu The application of the principles of Trigonometry to the 

measurement of portion of the earth's surface, ;>7i». 

" A branch ofpraotioa] science, 
Syllogism V form of statin.: the connection which may axial 

fur the purpo onimr, between three proposi- 

tions, 1". 

* A formula for ascertammg what may be predicated — 

J low it accomplishes tin-, 4 1. 
" not meant by Aristotle to he the form in which arguments 

should always 1. 53. 

" not a distinct kind of argument, 5 1 

" an argument stated at full length, 56. 

" Symbols QSed for the teimfl of, 56. 

" Rules for examining syllogisms, 07. 

" has three and only three terms, 67. 

" " " " " " propositions, 67. 

" test of deductive reasoning, 72, 99, 307. 

Symbols The letters which denote quantities, and the signs which 

indicate operations, 87, 93, 296. 

" used for the terms of a syllogism, 56. 

" Advantages of, 57. 

" Validity of the argument still evident, 58. 

" Truths inferred by means of, true of all things, '277. 

" regarded as things 27S. 

** Two classes of, in analysis, 296 

* Abstract and concrete quantity represented by, 321. 



INDEX. 373 



Synthesis The process of first considering the elements separately, 

then combining them, and ascertaining the results of 
combination, Sections 89, 327. 

Synthetical form, for what best adapted, 71, 89. 

Tables of Denominate Numbers, fractions occur five times in, 190. 

Technical Particular and limited sense, 91. 

Term Is an act of apprehension expressed in words, 15. 

" A singular term denotes but a single individual, 15. 

* A common denotes any individual of a whole class, 15. 

" " aifords the means of classification, 16. 

Nature of, 20. 
" " No real thing corresponding to, 20. 

M " Why applicable to several individuals, 20. 

Major Term — The predicate of the conclusion, 40. 
Minor Term — The subject of the conclusion, 40. 
Middle Term — The common term of the two premises, 40. 
Distributed — A term is distributed when it stands for all 
its significates, 61. 
'* not distributed — When it stands for a part of its sig- 

nificates only, 61. 

Terms Two of the three parts of a proposition, 38. 

" The antecedent and consequent of a proportion, 164, 267. 

" should always be used in the same sense, 170, 205. 

Text-Book Should be an aid to the teacher in imparting instruction, 

and to the learner in acquiring knowledge, 209. 

Thickness . . . . ; A dimension of space, E 2. 

Third Arithmetic, Principles contained in, and method of construction, 

231-236. 
Time, Measure of, its units and scale, 148. 
Topography, Its uses, 360. 

Trapezoid A quadrilateral, having two sides parallel, 242. 

Triangle A portion of a plane bounded by three straight lines, 241. 

" The simplest plane figure, 241. 

Different kinds of, 241. 
r regarded as a genus, 256. 

Trigonometry ....An application of the principles of Arithmetic, Algebra, 
and Geometry to the detenriination of the sides ana 
angles of triangles, 378. 
a Plane and Spherical, 378. 



374 INDEX. 



Troy Weight, Its units and scale, Section 137. 

Truth An exact accordance with what has been, is, or shal) 

be, 24. 
" Two methods of ascertaining, 24. 

" is inference from facts or other truths, 2-1, 25. 

" regarded as a species, 25. 

" How inferred from facts, 26. 

" A true proposition, 36. 

Truths Intuitive or Self-evident — Are such as become known 

by considering all the facts on which they depend, an 1 
apprehending the relations of those facta at the .-rue 
time, and by the same act by which we apprehend the 
facts themselves, 27. 
«• Logical — Those inferred from numerous and complicated 

facts ; and also, truths inferred from truths, 28. 
of Geometry, 237. 
Three classes of, 2 
" Demonstrative, 2 '.) 7. 

Unit fixed by the plare of the figure, 127. 

of the fraction. 100, 101. 
" of the expression, 160. 

Unities Advantages of the system of, 150-154. 

UNIT OF MEASURE. .The standard for measurement, 94. 
" for lines, surfaces, solids, 210. 

" only basis for e<timatin^ quantity, 251. 

Unit one A single tiling, 104. 

All numbers come from, 108, 109, 132, 150. 
" Method of impressing its values, 133. 

■ Three kinds of operations performed upon, 182-186. 

Units, Abstract or simple, 111, 132. 

" Denominate or Concrete, 111. 

" of currency, 132. 

of weight, 132. 
" of measure, 132, 139, 249. 

" of length, 140. 

" of surface, 141. 

" Duodecimal, 142. 

" of solidity, 145. 

u Fractional, 155, 185. 



INDEX. 375 



Unity — Unit Any thing regarded as a whole, Sections 109, 110 

Universal Proposition, 62. 

Utility and Progress, leading ideas, page 11. 

Variables Quantities which undergo certain changes of value, the 

laws of which are indicated by the algebraic expres- 
sions into which they enter, 282, 283, 313. 
" represented by the final letters of the alphabet, 284. 

Variations, Theory of, 285. 

Varying Scales, Units increasing by, 131, 183. 

Velocity known by measurement, 95. 

Weight known by measurement, 95. 

" A, should be exhibited to give ideas of numbers, 13S. 
Standard for, 254. 

Whateley, Archbishop, his views of logic, 72. 

Words, Definition of, 113. 

" expressing results of combinations, 193-197. 

* Double or incomplete sense of, 322. 

Zero The limit of a decreasing quantity, 302-306 



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to the educational public but to the reading community at large, that thousands of 
copies of the Fourth and Fifth Headers have found their way into public and private 
libraries throughout the country, -where they are in constant use as manuals of liter- 
ature, for reference as well as perusal. 

7. ARRANGEMENT. The exercises are so arranged as to present constantly al- 

nctice in the different styles of composition, while observing a definite 
plan of progression or gradation throughout the whole. In the higher books the ar- 
ticles are placed in formal s lassified topically, thus concentrating the in- 
st and inculcating a principle of association likely to prove valuable in subsequent 
general reading. 

8. NOTES AND BIOGRAPHICAL SKETCHES. These are full and adequate 

I .esent in pleasing style the history of 
every author laid under contribution. 

9. ILLUSTRATIONS. These are plentiful, almost profuse, and of the highest 
character of art They are found in every volume of the series as far as and including 
the Third Reader. 

10. THE GRADATION is perfect. Each volume overlaps its companion pre- 
ceding or following in the series, so that the scholar, in passing from one to another, 
is barely conscious, save by the presence of the new book, of the transition. 

11. THE PRICE is reasonable. The books were not trimmed to the minimum 
of size in order that the publishers might be able to denominate them Wk the cheapest 
in the market," but were made large enough d suffice for the grade indi- 
cated by the respective numbers. Thus the child is not compelled to go over his First 
Reader twice, or be driven into the Second before he is prepared for it. The compe- 
tent teachers who compiled the series made each volume just what it should be. leav- 
ing it for their brethren who should use the books to decide what constitutes true 
cheapness. A glance over the books will s„v ne that the same amount of 
matter is nowhere furnished at a price more reasonable. Besides which another con- 
sideration enters into the question of relative economy, namely, the 

12. BINDING. By the use of a material and process known only to themselves, 
in common with all the publications of this house, the National Readers are warranted 
to out-last any with which they may be compared— the ratio of relative durability be- 
ing ia their favor as two to one. 



The National Se?ies of Standard School-Books. 

SCHOOL-ROOM CARDS, 

To Accompany the National Headers. 
Eureka Alphabet Tablet *i 50 

Presents the alphabet upon the "Word Method System, by which the 
child will learn the alphabet in nine days, and make no small" progress in 
reading and spelling in the same time. 

National School Tablets, 10 Nos *7 so 

Embrace reading and conversational exercises, object and moral les- 
sons, form, color, &c. A complete set of these large and elegantly illus- 
trated Cards will embellish the school-room more than any other article 
of furniture. 



READING. 



Fowle's Bible Reader $1 00 

The narrative portions of the P.ible, chronologically and topically ar- 
ranged, judiciously combined with selections from the Psalms, 1'rov- 
and other portions which inculcate important moral lessons or the | I 
troths of Christianity. The embarrassment :md difficulty of r. ■adin.- 
Biblfl itself, by course, as a class exercise, are obviated, and its use made 
feasible, by this means. 

North Carolina First Reader 50 

North Carolina Second Reader 75 

North Carolina Third Reader l oo 

Prepared expressly for the schools of this State, by C. IT. "Wiley. Super- 
intendent of Common Schools, and F. M. Hubbard, Professor of Litera- 
ature in the State University. 

Parker's Rhetorical Reader l oo 

Designed to familiarize Readers with the pauses and other marks in 
general use, and lead them to the practice of modulation and inflection of 
the voice. 

Introductory Lessons in Reading and Elo- 
cution 75 

Of similar character to the foregoing, for less advanced classes. 

High School Literature l 50 

Admirable selections from a long list of the world's best writers, for ex- 
ercise in reading, oratory, and composition. Speeches, dialogues, and 
model letters represent the latter department. 

5 



The National Series of Standard School-lBooks. 

ort ho g rap hy; 

smith's" series 

Supplies a speller for every class in graded schools, and comprises the most com* 
* plete and excellent treatise on English Orthography and its companion 
branches extant. 

i. Smith's Little Speller $20 

First Round in the Ladder of Learning. 

2. Smith's Juvenile Definer 45 

Lessons composed of familiar words grouped with reference to similar 
signification or use, and correctly spelled, accented, and defined. 

3. Smith's Grammar-School Speller .... 

Familiar words, grouped with reference to the sameness of s~>und of syl- 
lables differently spelled. Also definitions, complete rules for spelling and 
formation of derivatives, and exercises in false orthography. 

4. Smith's Speller and Definer's Manual • 

A complete School Dictionary containing 14,000 words, with various 
other useful matter in the way of Rules and Exercises. 

5- Smith's Hand-Book of Etymology • . 1 25 

The first and only Etymology to recognize the Anglo-Saxon our mother 
tongue; containing also full lists of derivatives from the Latin, Greek, 
Gaelic, Swedish, Norman, <&c., &c ; being, in fact, a complete etymology 
of the language for schools. 

Sherwood's Writing Speller 15 

Sherwood's Speller and Definer 15 

Sherwood's Speller and Pronouncer ... 15 

The Writing Speller consists of properly ruled and numbered blanks 
to receive the words dictated by the teacher, with space for remarks and 
corrections. The other volumes may be used for the dictation or ordinary 
class exercises. 

Price's English Speller *15 

A complete spelling-book for all grades, containing more matter than 
"■Webster," manufactured in superior style, and sold at a lower price — 
consequently the cheapest speller extant. 

Norlhend's Dictation Exercises 63 



50 
90 



Embracing valuable information on a thousand topics, communicated 
in such a manner as at once to relieve the exercise of spelling of its usual 
tedium, and combine it with instruction of a general character calculated 
to profit and amuse. 



25 



Wright's Analytical Orthography .... 

This standard work is popular, because it teaches the elementary sounds 
|n a plain and philosophical manner, and presents orthography and or- 
thoepy in an easy, uniform system of analysis or parsing. 

Fowle's False Orthography ■ ..... 45 

Exercises for correction. 

Page's Normal Chart ......... *3 75 

The elementary sounds of the language for the school-room walls. 

6 



The *Yatio?ial Series o/ Standard School- '/Jvo&s. 



ENG LISH GRAM MAR. 

CLARK'S DIAGRAM SYSTEM. 

Clark's First Lessons in Grammar . . . 50 

Clark's English Grammar l oo 

Clark's Key to English Grammar ... 60 
Clark's Analysis of the English Language - go 
Clark's Grammatical Chart 4 oj 

The theory and practice of teaching grammar in American schools is 
meeting with a thorough revolution from the use of this system. While 
the old methods offer proficiency to the pupil only after much weary 
plodding and dull memorizing, this affords from the inception the ad- 
vantage of practical Object Teaching^ addressing the eye by means of il- 
lustrative figures; furnishes association to the memory, its most power- 
ful aid, and diverts the pupil hy taxing his Ingenuity. Teacheri who are 
using Clark's Grammar uniformly testify that they and their pupils find 
it the most interesting study of the school course. 

Like all great and radical Improvements, the system naturallv met at 
first with much unreasonable opposition. It has" not onl y outlived the 
greater part of this opposition, but finds many of its w ar m e at admirers 
among those, who could not at first tolerate so radical an innovation. All 
it wants is an impartial trial, to convince the most skeptical of its merit. 
No one who has fairly and intelligently tested it in the school-room has 
ever been known to go back to tin: old method. A great success is al- 
ready established, and it is easy to prophecy thai the day is not flu 
taut when it will be the orilygyctcm qfteacmng English Grammar , As 
the syhii:m is copyrighted, no other text-book's can' appropriate this ob- 
vious and great Improvement 

Welch's Analysis of the English Sentence • i 10 

Remarkable for its new and simple classification, its method of treat- 
ing connectives, its explanations of the idioms and constructive laws of 
the language, &c. 



ETYMOLOGY. 



Smith's Complete Etymology, l ^ 

Containing the Anglo-Saxon, French, Dutch, German, Welsh, Danish, 
Gothic, Swedish, Gaelic, Italian, Latin, and Greek Roots, and the English 
words derived therefrom accurately spelled, accented, and defined. 

The Topical Lexicon, 15: 

This work is a School Dictionary, nn Etymology, a compilation of syn- 
onyms, and a manual of general information. It differs from the ordinary 
lexicon in being arranged by topics instead of the letters of the alphabet, 
thus realizing the apparent paradox of a "Reliable Dictionary." An 
unusually valuable school-book. 

7 



The JYalional Series of Standard Sc?iool-!Book$* 



GEOGRAPHY. 



TIIE 

NATIONAL GEOGRAPHICAL SYSTEM. 



I. Monteith's First Lessons in Geography, % 35 

II. Monteith's Introduction to the Manual, • 65 

III. Monteith's New Manual of Geography, • l oo 

IV. Monteith's Physical & Intermediate Geog. i 75 

V. McNally's System of Geography, • ■ . l 88 

The only complete course of geographical instruction. Its circulation 
is almost universal — its merits patent. A few of the elements of its popu- 
larity are found in the following points of excellence. 



1, PRACTICAL OBJECT TEACHING. The infant scholar is first introduced 
to a picture whence he may derive notions of the shape of the earth, the phenomena 
of day and night, the distribution of land and water, and the great natural divisions, 
which mere words would fail entirely to convey to the untutored mind. Other pic- 
tures follow on the same plan, and the child's mind is called upon to grasp no idea 
without the aid of a pictorial illustration. Carried on to the higher books, this system 
culminates in No. 4, where such matters as climates, ocean currents, the winds, pecu- 
liarities of the earth's crust, clouds and rain, are pictorially explained and rendered 
apparent to the most obtuse. The illustrations used for this purpose belong to the 
highest grade of art, 

2. CLEAR, BEAUTIFUL, AND CORRECT MAPS. In the lower numbers 
the maps avoid unnecessary detail, while respectively progressive, and affording the 
pupil new matter for acquisition each time he approaches in the constantly enlarging 
circle the point of coincidence with previous lessons in the more elementary books. 
In No. 4, the maps embrace many new and striking features. One of the most 
effective of these is the new plan for displaying on each map the relative sizes of 
countries not represented, thus obviating much confusion which has arisen from the 
necessity of presenting maps in the same atlas drawn on different 6cales. The maps 
of No. 5 have long been celebrated for their superior beauty and completeness. This 
Is the only school-book in which the attempt to make a complete atlas also clear and 
distinct, has l>2en successful. The map coloring throughout the series is also notice- 
able. Delicate and subdued tints take the place of the startling glare of inharmonious 
colors which too frequently in such treatises dazzle the eyes, distract the attention, 
and serve to overwhelm the names of towns and the natnraJ features of the landscape. 

8 



The National Series of Standard Sc/iool-2?oo£s. 



GEOGRAPHY-Continued 

3. THE VARIETY OF MAP EXERCISE, Starting each time from a different 
basis, the pupil in many instances approaches the same fact no less than gix times, 
thus indelibly impressing it upon his memory. At the same time this system is mt 
allowed to become wearisome — the extent of exercise on each subject being graduated 
by its relative importance or difficulty of acquisition. 

4. THE CHARACTER AND ARRANGEMENT OF THE DESCRIPTIVE 
TEXT* The cream of the science has been carefully culled, unimportant matter re 
|ccted, elaboration avoided, and a brief and concise manner of presentation cultivated 
Tho orderly consideration of topics has contributed greatly to simplicity. Due atten 
lion is paid to the facts in history and astronomy which are inseparably connected 
with, and important to the proper understanding of geography — and such only at* 
admitted on any terms. In a word, the National System teaches geography as a 
science, pure, simple, and exhaustive, 

5. ALWAYS UP TO THE TIMES. The authors of these books, editorially 
speaking, never 6leep. No change occurs in the boundaries of countries, or of coun- 
ties, no new discovery is made, or railroad built, that is not at once noted and re- 
corded, and the next edition of each volume carries to every school-room the new or- 
der of things. 

6. SUPERIOR GRADATION. This is the only series which furnishes an avail- 
able volume for every possible class in graded schools. It is not contemplated that ■ 
pupil must necessarily go through every TOloBM in succession to attain proficiency. 
On the COtttrary, (WO will suffice. DuUJbm are advised ; and if the course will admit, 
the whole series should be pursued. At all events, the bookl are at baud for selection, 
and every teacher, of every . :.d among M vUtd to his class. 
The best combination for those who wish to abridge the course consists of Nos. 1, 3, 
and 5, or where children aro somewhat advanced In other studies when they com- 
mence geography, Nos. 2, 3, and B, Where but tiro books are admissible, Nos. 2 and 
4, or Nos. 3 and 5, are recommended. 

7. FORM OP THE VOLUMES AND MECHANICAL EXECUTION. The 

maps and text are no longer unnaturally divorced in aocordauee With the timo-hoa- 
ored practice of making text-books on this Subject as inconvenient v e as 

possible On the contrary, all map questions are to be found on the page opposite the 
map itself, and each book is complete in one volume. The mechanical execution is 
unrivalled. Paper and printing are everything that could be desired, and the bind- 
ing is— A. S. Barnes and Company's. 



Ripley's Map Drawing $* 25 

This system adopts the circle as its basis, abandoning the processes by 
triangulation, the square, parallels, aud meridians, &c, which havo been 
proved not feasible or natural iu the development of this science. Sue- 
cess seems to indicate that the circle l< has it." 

National Outline Maps 

Foi the school-room walls. In preparation. 

9 



The JYali07ial Series of Standard School-Books. 

MATHEMATICS. 

M¥I!F NATIONAL 6011111 

ARITHMETIC. 

1. Davies' Primary Arithmetic . . $ 25 

2. Davies' Intellectual Arithmetic 40 

3. Davies' Elements of Written Arithmetic 50 

4. Davies' Practical Arithmetic 1 00 

Key to Practical Arithmetic *1 00 

5. Davies' University Arithmetic 1 50 

Key to University Arithmetic *1 50 

ALGEBRA. 

1. Davies' New Elementary Algebra 1 25 

Key to Elementary Algebra *1 25 

2. Davies' University Algebra 1 60 

Key to University Algebra *i 6C 

3. Davies' Bourdon's Algebra 2 25 

Key to Bourdon's Algebra *2 25 

GEOMETRY, 

1, Davies' Elementary Geometry and Trigonometry . l 40 

2, Davies' Legendre's Geometry 2 25 

3. Davies' Analytical Geometry and Calculus . . . . . 2 50 

4. Davies' Descriptive Geometry 2 75 

MENSURATION. 

1. Davies' Practical Mathematics and Mensuration . . 1 40 

2. Davies' Surveying and Navigation 2 50 

3. Davies' Shades, Shadows, and Perspective . . . . 3 73 

MATHEMATICAL SCIENCE. 

Davies' Grammar of Arithmetic * 50 

Davies' Outlines of Mathematical Science .... *1 00 
Davies' Logic and Utility of Mathematics .... M 50 
Davies & Peck's Dictionary of Mathematics . .*3 75 

10 



The National Series of Standard School-lBooks. 

DAVIES' NATIONAL COUBSE of MATHEMATICS. 

ITS RECORD. 

In claiming for this series the first place among American text-bookr,, of whatever, 
class, the Publishers appeal to the magnificent record which its volumes have earned/ 
during the thirty-five years of Dr. Charles Davies 1 mathematical labors. The unre- 
mitting exertions of a life-time have placed the modern series ou the same proud emi- 
nence among competitors that each of its predecessors has successively enjoyed in a 
(course of constantly improved editions, now rounded to their perfect fruition — for it 
aeems indeed that this science is susceptible of no further demonstration. 
• During the period alluded to, many authors and editors in this department have 
Btarted into public notice, and by borrowing ideas and processes original with Dr. 
Davies, have enjoyed a brief popularity, but are now almost unknown. Many of the 
series of to-day, built upon a similar basis, and described as "modern books, *' aro 
destined to a similar fate ; while the most far-fleeing eye will find it difficult to fix the 
time, on the basis of any data afforded by their past history, when these books will 
cease to increase and prosper, and fix a still firmer hold on the affection of every 
educated American. 

One cause of this unparalleled popularity is found In the fact that the enterprise of 
the author did not cease with the original completion of his books. Always a practi- 
cal teacher, he has incorporated in his text-books from time to tinn- the advair 
of every improvement in methods of teaching, and every advance [n science. During 
nil the years in which he has been laboring, he constantly submitted his own theories 
and those of others to the practical test of the class-room — approving, rejecting, or 
modifying them as the experience thus obtained might suggest. In this way In- has 
been able to produce an almost perfect series of els I which every depart- 

ment of mathematics has received minute and exhaustive attention, 

Nor has he yet retired from the field* Still In the prime of life, and enjoying a rlpo 
experience which no other living mathematician or teacher can emulate, his pen is 
ever ready to carry on the good work, as the progress of science may demand. Wit- 
ness his recent exposition of the "Metric System," whi 1 the official en- 
dorsement of Congress, by its Committee on Uniform Weights and Measures. 

Davies* System is the acknowledged National BCAXBtASD POl am United 
States, for the following reasons : — 

1st. It is the- basis of instruction in the great national schools at West Point and 
Annapolis. 

2d. It has received the quasi endorsement of the National Congress. 

3d. It is exclusively used in the public schools of the National Capital. 

4th. The officials of the Government use it as authority in all cases involving mathe- 
matical questions. 

6th. Our great soldiers and sailors commanding the national armies and navies 
were educated in this system. So have been a majority of eminent scientists in thlfl 
country. All these refer to " Davies" as authority. 

( 6th. A larger number of American citizens have received their education from this 
than from any other series. 

Tth. The series has a larger circulation throughout the whole country than any 
other, being extensively used in every State in the Union, 

U 



27ie National Series of Standa?x2 Sc?iool-2$ooks. 

MATHEMATICS-Continued. 

ARITHMETICAL EXAMPLES. 

Reuck's Examples in Denominate Numbers % so 
Reuck's Examples in Arithmetic .... l oo 

These volumes differ from the ordinary arithmetic in their peculiarly 
"practical character. They are composed mainly of examples, and afford 
the most severe and thorough discipline for the mind. While a book 
\rhich should contain a complete treatise of theory and practice would be 
too cumbersome for cvery-day use, the insufficiency of practical examples 
has been a source of complaint. 

HIGHER MATHEMATICS. 

Church's Elements of Calculus 2 50 

Church's Analytical Geometry 2 50 

Church's Descriptive Geometry, with Shades, 

Shadows, and Perspective 4 50 

These volumes constitute the ""West Point Course 1 ' in their several 
departments. 

Courtenay's Elements of Calculus .... 3 25 

A work especially popular at the South. 

Hackley's Trigonometry 3 oo 

"With applications to navigation and surveying, nautical and practical 
geometry and geodesy, and logarithmic, trigonometrical, and nautical 
tables. 

THE METRIC SYSTEM. 

The International System of Uniform Weights and Measures must hereafter be 
tanght in all common-schools. Professor Charles Davies is the official exponent of 
the system, as indicated by the following resolutions, adopted by the Committee of the 
House of Representatives, on a " Uniform System of Coinage, Weights, and Measures," 
February 2, 1S67 :— 

Resolved, That this committee has observed -with gratification the efforts made by 
the editors and publishers of several mathematical works, designed for the use of com- 
mon-schools and other institutions of learning, to introduce the Metric System of 
Weights and Measures, as authorized by Congress, into the system of instruction of 
the youth of the United States, in its various departments ; and, in order to extend 
further the knowledge of its advantages, alike in public education and in general use 
by the people, 

Be it further resolved, That Professor Charles Davies, LL.D., of the State of Ne^r 
York, be requested to confer with superintendents of public instruction, and teachen 
of schools, and others interested in a reform of the present incongruous system, and, 
by lectures and addresses, to promote its general introduction and use. 

The official version of the Metric System, as prepared by Dr. Davies, may be found 
In the Written, Practical, aud University Arithmetics of the Mathematical Series, and 
la also published separately, price postpaid. Jive cents. 

12 



27ie National Series oj Standai*d School-Hooks. 

HISTORY. 



Monteith's Youth's History, $75 

A History of the United States for beginners. It is arranged upon the 
catechetical plan, with illustrative maps and engravings, review questions, 
dates iti parentheses (that their study may fee optional with the youn_-> r 
Class of learners ). and interesting Biographical Sketches of all pt.-rsons 

who have been prominently identified with the history of our country. 

Willard's United States, School edition, . . . l at 

Do. do. University edition, . 2 25 

The plon of this standard work is chronologically exhibited in front of 
the title-page ; the Maps and Sketches are found useful assistants to 4ba 
memory, and dates, usually so difficult to remember, arc so systematically 
arranged as in a great degree to difficulty. Candor, Impar- 

tiality, and accuracy, are the distinguishing features of the narrattvcj 
portion. 

Willard's Universal History, 2 25 

The mo-t valuable features of the "1 'lured in 

this. The peculiarities Of tfa • work are its great I and tin- 

prominence given tn the chronological ord The margin 

marks each BUCOeSBive < ra with . BO that the pupil re- 

tains not only the erent but its time, and thus fixes the order of history 
firmly and usefully in lis i .Willard's books tantiy 

revised, and at ell times written up \\> unbraeo important historical 

events of i 

Berard's History of England, 1* 

By an authoress well known i~<>r I f her History of the Doited 

States. The Social life ol t! felicitously i'. 

as iu fact, with the civil an.; military tra realm. 

Ricord's History of Rome, i 2fi 

Possesses the charm of an attractiTo romance. The tables with which 
this history abounds are introduced i.i BUCh a way as m.: t < . d« rtive the 
Inexperienced, while adding materi Jly to the value of the work as a reli- 
able index to the character and in 1 as the history of the 
Roman people 

Hanna's Bible History, i 25 

The only compendium of Bible narrative which affords a connected and 
chronological view of the important events there recorded, divested of all 
superfluous detail. 

Alison's History of Europe 2 

An abridgment for Schools, by Gould, of this great standard work, 
covering the eventful period from A. 1). 17S9 to lSi5, being mainly a his- 
tory of the career of Napoleon. 

Marsh's Ecclesiastical History, 1 88 

Questions to ditto, 75 

Affording the History of the Church in nil ages, with accounts of the 
pagan world during Biblical periods, and the character, rise, and progress 
of all Religions, as well as the various sects of the worshipers of Christ. 
The work is entirely non-sectarian, though strictly catholic. 

13 



The National Series of Standard Scfoool-!&ooks. 

PENMANSHIP. 

■» <Q» <♦ 

Beers' System of Progressive Penmanship. 

Per dozen $2 50 

This " round hand " system of penmanship in twelve numbers com- 
mends itself by its simplicity and thoroughness. The first four numbers 
are primary books. Nee. 5 to 7, advanced books for boys. Nos. 8 to 10 
advanced books for girls. Nos. 11 and 12, ornamental penmanship. 
These books are printed from steel plates (engraved by McLees), and are 
unexcelled in mechanical execution. Large quantities are aunually sold. 

Beers' Slated Copy Slips, per set *50 

All beginners should practice, for a few weeks, slate exercises, familiar- 
izing them with the form of the letters, the motions of the hand and arm, 
&c., &c. These copy slips, 32 in number, supply all tho copies found in a 
complete series of writing-books, at a trifling cost. 

Fulton & Eastman's Copy Books, per dozen l 50 

A series for the economical, — complete in three numbers. (1) Elemen- 
tary Exercises : (2) Gentlemen's Hand : (3) Ladies' Hand. 

Fulton & Eastman's Chirographic Charts, 

2 Nos., per set *5 00 

To embellish the school-room walls, and furnish class exercise in the 
elements of Penmanship. 

DRAWING. 

Clark's Elements of Drawing l 00 

Containing full instructions, with appropriate designs and copies for a 
complete course in this graceful art, from the first rudiments of outline to 
the finished sketches of landscape and scenery. 

Fowle's Linear and Perspective Drawing 60 

For the cultivation of the eye and hand, with copious illustrations and 
directions which will enable the unskilled teacher to learn the art himself 
while instructing his pupils. 

Monk's Drawing Books— Six Numbers, each, .* 40 

A series of progressive Drawing Eooks, presenting copy and blank on 
opposite pages. The copies are fac-similes of the best imported litho- 
graphs, the originals of which cost from 50 cents to $1.50 each in the 
print-stores. Each book contains eleven large patterns. No. 1. — Ele- 
mentary studies ; No. 2.— Studies of Foliage; No. 3.— Landscapes ; No. 
4.— Animals, I. ; No. 5.— Animals, II. ; No. C— Marine Views, &c. 

Ripley's Map Drawing l 25 

One of the most efficient aids to the acquirement of a knowledge of 
geography is the practice of map drawing. It is useful for the same rea- 
son that the best exercise in orthography is the writing of difficult words. 
Sight comes to the aid of hearing, and a double impression is produced 
upon the memory. Knowledge becomes less mechanical and more intui- 
tive. The student who has sketched the outlines of a country, and dotted 
the important places, is little likely to forget either. The impression pro- 
duced may be compared to that of a traveler who has been over the 
ground — while more comprehensive and accurate in detail. 

M 



The Natio?ial Series of Standard School- Itooks. 

ELOCUTION. 



Northend's Little Orator *60 

Contains simple and attractive pieces in prose and poetry, adapted to 
the capacity of children under twelve years of age. 

Northend's National Orator *i 10 

Ahout one hundred and seventy choice pieces happily arranged. The 
design of the author in making the selection has been to cultivate versa- 
tility of expression. 

Northend's Entertaining Dialogues • • . .*l 10 

Extracts eminently adapted to cultivate the dramatic faculties, as well 
as entertain an audit 

Zachos' Analytic Elocution l 25 

All departments of elocution — such .-is the ai I Hie 

sentence, phonology, rhythm, ex] r arranged 

for Instruction In classes, Illustrated by copious imnnplti 

Sherwood's Self Culture l 25 

Self culture in reading, speaking, and valuable 

treatise to those who would ; i icompliahn 



B OOK-KEEPING. 



Smith & Martin's Book-keeping 1 10 

Blanks to ditto *60 

This work is by ■ practical teacher and a practical book-keeper. It is 
of a thoroughly popular class, and will be welcom «1 by who 

lores te Bee theory and pi tbined in i . and 

methodical form 

The Single Entry portion is well adapted to supply a want felt in nearly 
all other treatises, which seem to be prepared mainly for I 
wholesale merchants, leaving retailers, mechanics, farmers, &c, who 
transact the greater portion of the business of the country, without a 
guide. The work is also commended on this account for general use in 
Young Ladies 1 seminaries, where a thorough grounding in the simpler form 
of accounts will be invaluable to Ihe future housekeepers of the nation. 

The treatise on Double. Entry Book-keeping combines all the advan- 
tages of the most recent methods, with the utmost simplicity of applica- 
tion, thus affording the pupil all the advantages of actual experience in 
the counting-house, and giving a clear comprehension of the entire sub- 
ject through a judicious course of mercantile transactions. 

The shape of the hook is such that the transactions can be presented as 
in actual practice; and th>^ simplified form of Blanks, three in number, 
adds greatly to the case experienced in acquiring the science. 

15 



The National Series of Standard Sckool-JBook* 

NATURAL SCIENCE. 



D 



FAMILIAR SCIENCE 
Norton & Porter's First Book of Science, • $l 50 

By eminent Professors of Yale College. Contains the principles of 
Natural Philosophy. Astronomy, Chemistry, Physiology, and Geology. 
Arrauged on the Catechetical plan for primary classes and beginners. 

Chambers' Treasury of Knowledge, ■ • ■ l 10 

Progressive lessons upon— yzrsf, common things which lie most imme- 
diately around us, and first attract the attention of the young mind; 
second, common objects from the Mineral, Animal, and Vegetable king- 
doms, manufactured articles, and miscellaneous substances ; third, a sys- 
tematic view of Nature under the various sciences. May be used as a 
Reader or Text-Book. 

NATURAL PHILOSOPHY. 
Norton's First Book in Natural Philosophy, 90 

By Prof. Nobxox, of \ I for beginners; profusely 

illustrated, and arranged on the Ua : Ian. 

Peck's Ganot's Course of Nat. Philosophy, l 71 

The standard text-book of Franc, Americanized and popularized by 
Prof. Peck, of Colombia College. The most magnificent system of illus- 
tration ever ad >pte 1 in an American school-book is here found. For 

intermediate cl.t 

Peck's Elements of Mechanics, 2 25 

A suitable introduction to Bartlett's higher treatises on Mechanical 
Philosophy, and adequate in itself for a complete academical course. 

Bartlett's Synthetic Mechanics, 3 75 

Bartlett's Analytical Mechanics, 5 50 

Bartlett's Acoustics and Optics, ..... 2 75 

A system of Collegiate Philosophy, by Prof. Bautlett, of ^Vest Point 
Military Academy. 

GEOLOGY. 
Page's Elements of Geology, 1 10 

A volume of Chambers' Educational Course. Practical, simple, and 
eminently calculated to make the study interesting. 

Emmon's Manual of Geology, 1 25 

The first Geologist of the country has here produced a work worthy of 
his reputation. The plan of presenting the subject is an obvious improve- 
ment on older methods. The department of Palaeontology receives espo- 
cial attention. -* p 



The National Se?*ies of Sta?idard School-Books. 

NATURAL SCIENCE-Continued. 

CHEMISTRY, 

Porter's First Book of Chemistry, . . . .% 90 
Porter's Principles of Chemistry, .... l 88 

The above are widely known as the productions of one of the most 
eminent scientific men of America. The extreme simplicity in the method 
of presenting the science, while .••■d, has excited uni- 
versal commendation. Appa performance of every 
experiment mentioned, may ; ublishera for a trifling sum. 
The -effort to popularize the it success. It is now within 
the reach of the poorest and 1 . '■;..• at once. 

Darby's Text-Book of Chemistry, . . . . 1 60 

Puii of matters comparatively 

foreign to II heat, light, bat usually allowed to 

engross too ma ks. 

Gregory's Organic Chemistry, 2 60 

Gregory's Inorganic Chemistry, 2 50 

The science exhaustively treated. For Colleges and medical students, 

Steele's Fourteen Weeks' Course, i 25 

A successful effort to reduce th rm, 

thereby makl i institutions of erery 
character The author's (elicit] in maktag t ho 

ace pre-eminently i i are pecnliai 

Chemical Apparatus, *H oo 

Oomprifling articles adequate for the practical illustration of the exper- 
iments la Porter's or fi 

BOTANY. 
Thinker's First Lessons in Botany, ... 40 

For children. T 1 with in faror 

of an easy and familiar Style adapted to the rnef. 

Wood's Object Lessons in Botany, . . . . i 40 

Wood's Intermediate Botany, a 25 

Wood's New Class-Book of Botany, ... .m 

The standard text-hooks of the United States in this department In 
style they are simple, popular, and lively; In arrangement, easy and nat- 
ural; in description, graphic and strictly exact. The Tables for Analysis 
are reduced to a perfect system. More are annually sold than of all others 
combined. 

Darby's Southern Botany, 2 oo 

Embracing general ^ructural and Physiological Botany, with vegetable 
products, and descriptions of Southern plants, and a complete Flora of 
the Southern States, 

17 



The National Series of Standard School-Books* 
NATURAL SCIENCE-Continued 

PHYSIOLOGY. 

Jams' Primary Physiology, $75 

Jarvis' Physiology and Laws of Health, • 1 50 

The only books extant which approach this subject with a proper view 
of the true object of teaching Physiology in schools, viz., that scholars 
may know how to take care of their own health. In bold contrast with 
the abstract Anatomies, which children learn as they would Greek or 
Latin (and forget as soon), to discipline the mind, are these text-book?, 
using the science as a secondary consideration, and only so far as is 
necessary for the comprehension of the laws of health. 

Hamilton's Vegetable & Animal Physiology, i 10 

The two branches of the science combined in one volume lead the stu- 
dent to a proper comprehension of the Analogies of Nature. 

ASTRONOMY. 
Willard's School Astronomy, 90 

By means of clear and attractive illustrations, addressing the eye in 
many cases by analogies, careful definitions of all necessary technical 
term's, a caieful avoidance of verbiage and unimportant matter, particular 
attention to analysis, and a general adoption of the simplest methods, 
Mrs. Willard has made the best and mosi attractive elementary Astron- 
omy extant. 

Mclntyre's Astronomy and the Globes, • 1 25 

A complete treatise for intermediate classes. Highly appr 

Bartlett's Spherical Astronomy, 4 50 

The West Point course, for advanced classes, with applications to th* 
current wants of Navigation, Geography, and Chronology. 

NATURAL HISTORY. 
Carl's Child's Book of Natural History, . . o so 

Illustrating the Animal, Vegetable, and Mineral Kingdoms, with appli- 
cation to the Arts. For beginners. Beautifully and copiously illustrated. 

ZOOLOGY, 
Chambers' Elements of Zoology, l 50 

A complete and comprehensive system of Zoologv, adapted for aca- 
demic instruction, presenting a systematic view of the Animal Kingdom 
as a portion of external Nature. 

i ~ s 

S E^~ It will be observed, that, iu the various departments of Natural Science, the 
!Natioxal Series is extremely rich. The mineral, animal, and vegetable kingdoms, 
matter, and the laws that govern it in all its forms, are here placed before the 
Student by those who have made its study a specialty and a life work. The -vrorks 
of Professors Peck, of Columbia College, Norton and Porter, of Yale, Bart- 
LETT, of West Point Military Academy. Emmets, of Williams, and State Geologist 
of New York and North Carolina, Wood, the botanist, and Jarvis, the eminent phy- 
sician, are esteemed indubitable authority in all that concerns their several specialties 

18 



2*he National Series of Standard School-Books. 

MODERN LANGUAGE. 

French and English Primer, $ 10 

German and English Primer, 10 

Spanish and English Primer, 10 

The names of common objects properly illustrated and arranged in easy 

Ledru's French Fables, 75 

Ledru's French Grammar, i oo 

Ledru's French Reader, i oo 

Th • author's I i se has enable l bin to present 1 1 1 *- o 

oughly practical text-books extant, in this branch. The system of pro- 
nunciation (by phonetic Ulnstratio I with thia author, and will 
commend itself to all \ ' :,,s their pnpili I 

i without the assistance ol ■ native 
tture is p euliarly valuable al wight" stir! 

The direct! ' nouns— eJ 

,bll ig-block md remark 

compel ml to the end proposed The cr 
the echo A-room Is invlt \a to thla exc tllenl seri is, irltb - 

Haskin's French and English First Book . 76 

Pi -s mta the Btriking feature of ■ limn 
mentary principles of the vernacular with tb l an image. 

This is the metuod wbiob th r naturallj 

instruction, and , i appUcati d to young 

pupils. 

Pujol's Complete French Class-Book, . . . 2 25 

Offers, i'i one volume, methodically arranged, s complete French co n 
— usually embraced In eeriei of from nVe to twelve books, includln • 
bulky and expensire Lexicon. Here are Chrami 
choi< From the best French authors, Bach braadi 

is thoroughly handle 1 ; and the student, baring diligently completed ibo 
course as prescribed, may consider himself, without further applicationi 
tm fait in the most polite and elegant language of modern tin 

Maurice-Poitevin's Grammaire Francaise, • l 09 

American Bchoola are el last supplied with an American edition of this 
famous text-book. Many of our best Institutions have tor years been pro- 
curing it from abroad rather than forego the advantages it offers. The 
policy of putting students who have acquired some proficiency from the 
ordinary text-books, into a Grammar written in the vernacular, can not 
be too lughly commended. It affords an opportunity for finish and review 
at once ; while embodying abundant practice of its own rules. 

Worman's Elementary German Grammar, • l 12 

A work of great merit Well calculated to ground tbe student in the 
elements of this language, become so important by the extensive settle- 
ment of Germans in tbis country. 

Willard's Historia de los Estados Unidos, • 2 oo 

The History of the United States, translated by Professors Tolon and 
Dr. Totsnob, will be found a valuable, instructive, and entertaining read- 
ing-book for Spanish classes. 

10 






T/ie National Series of Standard School-Books, 

THE CLASSICS. 



BROOKS' SERIES. 
Brooks' First Lessons in Latin $75 

Is composed of a legular series of inductive exercises for beginners, and 
is a Grammar, Reader, and Dictionary combined. 

Brooks' Historia Sacra, 75 

Selections from the Scriptures, in easy Latin, for lower classes. In- 
cludes copious notes and a Lexicon, with exercises in Grammar. 

Brooks' First Greek Lessons, 75 

Arranged on the same plan as the author's Latin Lessons. 

Brooks' Collectanea Evangelica, 75 

Selections from the four Gospels, in Greek, forming a connected history 
of the life of the Saviour, Notes and Lexicon. 

Brooks' Ross' Latin Grammar, 1 00 

A very popular text-book — much improved by Prof. Brooks 1 ex- 
perienced hand. 

Brooks' Viri Illustres Americae, l 50 

The lives of illustrious Americans, In Latin, with copious Illustrations, 
Notes, and Lexicon. The language is used in its purest type, while sim- 
ple, and adapted to the wants of students not yet prepared for the more 
difficult construction of classical authors. 

Brooks' Caesar's Commentaries, 1 50 

The handsomest and most complete annotated edition of any classical 
author ever published. The volume contains a map of Gaul, Life of Caesar 
in English, Tabic of Epochs, eleven full-page battle plaus, countless illus- 
trative engravings, an extended "Argument" prefacing each book, full 
Notes, References to Andrews & Stoddard's Latin Grammar, and a com- 
plete Lexicon— all furnished at as low price as any other edition. 

Brooks' Ovid's Metamorphoses 2 50 

Possesses all the advantages claimed for the preceding volumes-. All 
objectionable matter is carefully expurgated. 



Silber's Latin Course, 1 25 

A complete work, with Grammar, Reader, Notes, Lexicon, and Refer- 
ences to the three leading Latin Grammars — Andrews & Stoddard's, 
Bullions', and Harkness\ 

Dwight's Grecian and Roman Mythology. 

School edition, $1 25 ; University edition,*2 25 

A knowledge of the fables of antiquity, thus presented in a systematic 
form, is as indispensable to the student of general literature as to him 
who would peruse intelligently the classical authors. The mythological 
allusions so frequent in literature are readily understood with such a Key 

as tins. 

20 



2'he National Scries of Standard School - Dooks. 



LITERATURE. 



T H E O It Y 



Brookfield's First Book in Composition, • •$ 50 

Making the cultivation of this important art feasible for the Emallest 
child. By a new method, to Induce and stimulate thought. 

Boyd's Composition and Rhetoric, .... l 25 

This work furnishes all the aid that is needful OT can be desired in the 
various departments and styles of composition, hoth in prose and Terse. 

Day's Art of Rhetoric, I 2$ 

Noted for Of definition, clear limitation, and philosophical 

develepraenl of uubject; tfa ren to Invention, 

as a branch of Rhetoric, and the Dnequ yle. 

Boyd's Karnes' Criticism, l 75 

The beat edition of this standard - nt the study of which none 

may be considered proficient In B U I 

Walker's Rhyming Dictionary *J 25 

A complete lexicon of allowable rhymes, to aid the composer of ryth- 
mical matter. 

PRACTICE. 

Boyd's Milton's Paradise Lost, *i 25 

Boyd's Young's Night Thoughts, *i 25 

Boyd's Cowper's Task, Table Talk, &c, ■** 

Boyd's Thomson's Seasons, *i 2S 

Boyd's Pollok's Course of Time, *i 25 

Boyd's Lord Bacon's Essays, *i 7 5 

This series of annotated editions of great English writers, in prose and 
poetry, is designed for critical reading and parsing in schools. Prof. J. It. 
Boyd proves himself an editor of high capacity, and the works themselves 
need no encomium. As auxiliary to tho study of Composition and Rheto- 
ric these works have no equal. 

Pope's Essay on Man, *2o 

Pope's Homer's Iliad, *i 00 

The metrical translation of the great poet of antiquity, and the matchless 
» 4 Essay on the Nature and State of Man," by Alexjl*1>E3 Tors, afford 
superior escrcisc ia literature and parsing. 

21 



The National Series of Standard School~32ooks. 

MENTAL PHILOSOPHY. 



Mahan's Intellectual Philosophy $1 50 

The subject exhaustively considered. The author has evinced learn- 
ing, candor, and independent thinking. 

Mahan's Science of Logic 1 88 

A profound analysis of the laws of thought. The system possesses the 
merit of being intelligible and self-consistent. In addition to the author's 
carefully elaborated views, it embraces results attained by the ablest 
minds of Great Britain, Germany, and France, in this department. 

Boyd's Elements of Logic i oo 

A systematic and philosophic condensation of the subject, fortified with 
additions from "VVatts, Abercrombie, Whateley, &c 

Watts on the Mind 50 

The Improvement of the Mind, by Isaac Watts, is designed as a guide 
for the attainment of useful knowledge. As a text-book it is unparalleled ; 
and the discipline it affords cannot be too highly esteemed by the edu- 
cator. 

MORAL PHILOSOPHY. 

Alden's Text-Book of Ethics, 60 

For young pupils. To aid in systematizing the ethical teachings of the 
Bible, and point out the coincidences between the instructions of the 
sacred volume and the sound conclusions of reason. 

Willard's Morals for the Young, . . . .* 60 

Lessons in conversational style to inculcate the elements of moral phi- 
losophy. The study is made attractive by narratives and engravings. 

POLITICAL SCIENCE. 

1 j * > 

Young Citizen's Catechism 60 

Explaining the duties of District, Town, City, County, State, and 
United States Officers, with rules for parliamentary and commercial busi- 
ness — that which every future 4k sovereign' 1 ought to know, and so few are 
taught. 

Mansfield's Political Manual i 25 

This is a complete view of the theory and practice of the General and 
State Governments of the United States, designed as a text-book. The 
author is an esteemed and able professor of constitutional law. widely 
known for his sagacious utterances in matters of statecraft through the 
public press. Recent events teach with emphasis the vital necessity that 
the rising generation should comprehend the noble polity of the American 
government, that they may act intelligently when endowed with a voice 
in ft. 

22 



The JVallo?ial Series of Standa? % d School-Hooks. 



TEACHERS' AIDS. 



Brooks' School Manual of Devotion ... 60 

This volume contains daily devotional exercises, consisting of a hymn, 
selections of scripture for alternate reading by teacher and pupils, and a 
prayer. Its value for opening and closing school is apparent. 

Cleaveland's School Harmonist *75 

Contains appropriate tunes for each hymn in the "Manual of Devo- 
tion" described above. 

The Boy Soldier go 

Complete infantry tactics for schools, with illustrations, for the use of 
those who would introduce this pleasing relaxation from the confining 
duties of the desk. 

Welch's Object Lessons *75 

Invaluable for teachen of primary schools. Contains the best explana- 
tion of the Pestalocsian system. By its aid the proficiency of pupils and 
the. general interest of the school may be increased one hundred per cent 

Tracy's School Record *75 

To record attendance, deportment, and scholarship; containing also 
many useful tables and Suggestions to teachers, that are worth of them- 
selves the price of the book. 

Tracy's Pocket Record *co 

A portable edition of the School Record, without the tahles, Sec. 

Brooks' Teacher's Register * 90 

Presents at one view a record of attendance, recitations, and deport- 
ment for the whole term. 

Carter's Record and Roll-Book *i 50 

For large graded schools. 



National School Diary, per dozen *i 25 

weekly 
great 

23 



A little book of blank forms for weekly report of the standing of each 
scholar, from teacher to parent. A great convenience. 



The JV % atlo?ial 2'eache??* Z,ib7\i7y. 

THE 

TEACHERS LIBRARY. 



Too many teachers deem that the Normal School or Collegiate training -which they 
may have received before entering upon the active duties of their profession, fcs all- 
sufficient preparation for their responsible calling; and, thenceforth, the daily rootin€ 
of the school-room is to constitute the sole professional exercise that is incur; bent 
upon them. Never -was greater mistake. The teacher who is not proaiessive hin 
self, will never send forth scholars who are proficient. Those who hav« Deen tn# 
most successful, and whose names are honored among all the brotherhood, attribute 
their success more to personal magnetism — the spirit of "Come, boys!" not "Go, 
boys!" — than to any other agency which they bring to their task. The student 
teacher makes student scholars. 

The first question which the wide-awake teacher asks himself, after entering upon 
his duties and properly organizing his school, is, " What course for * ^/-improvement 
can I most advantageously pursue ? " and the answer which naturally presents itself— 
" Avail myself of the experience of those who have preceded me." This is easy, 
thanks to enterprising men, who have studied and labored, despaired and rejoiced, 
suffered and reaped their reward in the same paths upon which thousands of new foot- 
steps enter annually. With a whole-souled love of their profession, and regard for 
those who pursue it, they have committed their experience and its lessons to the press, 
so that " he who runs may read." 

In no way can teachers so fitly benefit themselves as in the perusal of these vol- 
umes, in which they should take a professional pride. Here are the records and re- 
sults of the life labor of those who have stood highest in the ranks, and who are pecu- 
liarly qualified for the task they undertake. 



Object Lessons-Welch ** 75 

This is a complete exposition of the popular modern system of M object- 
teaching," for teachers of primary classes. 

Theory and Practice of Teaching— Page •** 50 

This volume has, without doubt, been read by two hundred thousand 
teachers, and its popularity remains undiminished — large editions being 
exhausted yearly. It was the pioneer, as it is now the patriarch of 
professional works for teachers. 

The Graded School-Wells *i 25 

The proper way to organize graded schools is here illustrated. The 
author has availed himself of the best elements of the several systems 
prevalent in Boston. New York, Philadelphia, Cincinnati, St. Louis, and 
other cities. 

The Normal-Holbrook *i 50 

Carries a working school on its visit to teachers, showing the most ap- 
proved methods of teaching all the common branches, including the tech- 
nicalities, explanations, demonstrations, and definitions introductory and 
peculiar to each branch. 

The Teachers' Institute— Fowle *1 25 

This is a volume of suggestions inspired by the author's experience at 
Institutes, in the instruction of young teachers. A thousand points of in- 
terest to this class are most satisfactorilv dealt with. 

24' 



The National Teachers' Library. 

The Teacher and the Parent— Northend • 1*1 50 

A treatise upon common-school education, designed to lead teachers to 
Tiew their calling in its true light, and to stimulate them to fidelity. 

The Teachers' Assistant— Northend • • .*i so 

A natural continuation of the author's previous -work, more directly 
calculated for daily use in the administration of school discipline and in- 
struction. 

School Government— Jewell *i 50 

Full of advanced ideas on the subject which its title indicates. The cri- 
ticisms upon current theories of punishment and schemes of administra- 
tion have excited general attention and comment. 

Grammatical Diagrams— Jewell *i oo 

The diagram system of teaching gTMniMT explained, defended, and 
improved. The curious in literature, the searcher for truth, those inter- 
ested in new inventions, as well as the disciples of Prof. Clark, who would 
see their favorite theory fairly treated, all want this hook. There are 
many who would like to be made familiar with this system before risking 
its use in a class. The opportunity is hero aff >rded. 

The Complete Examiner— Stone *1 25 

Consists of a series of qiiettloafl OH every English branch of school and 
academic, instruction, wilh re f erence to ■ girei page or article of leading 
text-books where the answer may be found in full. Prepei 
teachers in eecariDg certificates, pnpile faa preparing for promotion, tad 
teachers in selecting review qoeaticnB. 

School Amusements— Root *1 50 

To assist teechen In making the school interesting, with hints upon the 
management of the school-room, Rnlea for military and gymnastic exer- 
cises are included. Illustrated by diagrams. 

Institute Lectures on Mental and Moral 

Culture-Bates *i M> 

These lectures, originally delivered before institutes, are based upon 
various topics of interest to the teacher. The volume is calculated to 
prepare the will, awaken the inquiry, and stimulate the thought of the 
zealous teacher. 

Method of Teachers' Institutes— Bates . . .* 75 

Sets forth the best method of conducting institutes, with a detailed ac- 
count of the object, organization, plan of instruction, and true theory of 
education on which such instruction should be based. 

History and Progress of Education • ■ .H 50 

The systems of education prevailing in all nations and ages, the gradual 
advance to the present time, and the bearing of the past upon the present 
in this regard, ar*. worthy of the careful investigation of all concerned in 
the cause, 

25 



The National Teachers' .Library. 



American Education— Mansfield $1 5C 

A treatise on the principles and elements of education, as practiced in 
this country, with ideas towards distinctive republican and Christian edu- 
cation. 

American Institutions— De Tocqueville . -*l 50 

A valuable index to the genius of our Government. 

Universal Education— Mayhew *l 50 

The subject is approached -with the clear, keen perception of one who 
has observed its necessity, and realized its feasibility and expediency 
alike. The redeeming and elevating power of improved common schools 
constitutes the inspiration of the volume. 

Higher Christian Education— Dwight ■ • *1 50 

A treatise on the principles and spirit, the modes, directions, and ra- 
suits of all true teaching ; showing that right education should appeal to 
every element of enthusiasm in the teacher's nature. 

Modern Philology— Dwight *l 75 

Important to the grammarian, and indispensable to the teacher of lan- 
guage, ancient or modern, who would afford his pupils the advantage of 
the analogy and association to be derived from an intelligent comparison 
of all languages and their history. 

Lectures on Natural History— Chadbourne * 75 

Affording many themes for oral instruction in this interesting science — 
especially in schools where it is not pursued as a class exercise. 

Outlines of Mathematical Science— Davies *1 oo 

A manual suggesting the best methods of presenting mathematical in- 
struction on the part of the teacher, with that comprehensive view of the 
whole which is necessary to the intelligent treatment of a part, in science. 

Logic & Utility of Mathematics— Davies ■ .*i 50 

An elaborate and lucid exposition of the principles which lie at the 
foundation of pure mathematics, with a highly ingenious application of 
their results to the development of the essential idea of the different 
branches of the science. 

Mathematical Dictionary— Davies & Peck . 3 75 

This cyclopsedia of mathematical science defines with completeness, 
precision, and accuracy, every technical term, thus constituting a populai 
treatise on each branch, and a general view of the whole subject. 

School Architecture— Barnard 2 25 

Attention is here called to the vital connection between a good school- 
house and a good school, with plans and specifications for securing the 
former in the most economical and satisfactorv manner. 

26 



National School Library* 



THE SCHOOL LIBRARY. 



The two elements of instruction and entertainment were never more happily com- 
bined than in this collection of standard hooks. Children and adults alike will hers 
find ample food for the mind, of the sort that is easily digested, while not degene- 
rating to the level of modern romance. 

LIBRARY OF LITERATURE. 
Milton's Paradise Lost Boyd's Illustrated Ed.$i Tfl 
Young's Night Thoughts .... do. . . l 75 
Cowper's Task, Table Talk, &c. • do. . . l 75 

Thomson's Seasons do. . . l 75 

Pollok's Course of Time .... do. . . l 75 

These great moral poemi ire known wherever the English language Is 

read, and arc regarded as models of the lust and purest literature. The 
hook;; are beautifully illustrated, and notes explain nil doubtful in 
IngS, and furnibh other matter ol" interest to the general reader. 

Lord Bacon's Essays, (Boyd's Edition.) ... l 75 

Another grand English classic, affording the highest example of purity 
in Language and style. 

The Iliad Of Homer. Translated by Pope. . . 1 00 

Those who are unable to road this greatest of ancient writers in the 
original, Should not fail to avail themselves of this metrical version by an 

eminent scholar and port. 

The Poets of Connecticut— Everest • • • • 1 75 

With the biographical sketches, this volume forms a complete history 
of the poetical literature of the State. 

The Son of a Genius— Hofland 75 

A juvenile classic which never wears out, and finds many interested 
readers in every generation of youth. 

Lady Willoughby i oo 

The diary of a wife and mother, An historical romance of the seven- 
teenth century. At once beautiful and pathetic, entertaining and in- 
structive. 

The Rhyming Dictionary— Walker • • • • l 25 

A serviceable manual to composers of rhythmical matter, being a com- 
plete index of allowable rhymes, 

27 



National School Library. 



LIBRARY OF REFERENCE. 
Home Cyclopaedia of Chronology ... .$2 25 

An Index to the sources of knowledge — a dictionary of dates. 

Home Cyclopaedia of Geography 2 25 

A complete gazetteer of the world. 

Home Cyclopaedia of Useful Arts 2 25 

Covering the principles and practice of modern scientific enterprise, 
with a record of important inventions in agriculture, architecture, do- 
mestic economy, engineering, machinery, manufactures, mining, photo- 
genic and telegraphic art, ifcc, &c. 

Home Cyclopaedia of Literature & Fine Arts 2 25 

A complete index to all terms employed in belles lettres, philosophy, 
theology, law, mythology, painting, music, sculpture, architecture, and all 
kindred arts. 



LIBRARY OF TRAVEL. 
Ship and Shore— Colton l 50 

In Madeira, Lisbon, and the Mediterranean Ocean. Illustrated. 

Land and Lee— Colton l 50 

In the Bosphorus and ^Egean. Illustrated. 

Sea and Sailor— Colton l 50 

Notes on France and Italy. Illustrated. 

Deck and Port— Colton l 50 

A cruise to California. Illustrated. 

Three Years in California— Colton .... l 50 

During the gold fever. Illustrated. 

These racy descriptions of travel are regarded as models in this 
department of literature. They are read by old and young with vast 
interest and profit. 

A Visit to Europe— Silliman, 2 vols 3 00 

A very spicy book of foreign travel. It brings every opportunity of the 
tourist to thr feet of the reader. 

28 



National School Zibra?y. 



TRAVEL-Continued. 

Life in the Sandwich Islands— Cheever • .$1 50 

The "heart of the Pacific, as it was and is, 11 shows most vividly the 
contrast between the depth of degradation and barbarism, and the light 
and liberty of civilization, so raDidly realized in these islands under the 
humanizing influence of the Christian religion. Illustrated. 

Peruvian Antiquities— Von Tschudi- - ■ . l 50 
Travels in Peru— Von Tschudi i so 

The first of these volumes affords whatever Information has been at- 
tained by travelers and men of science concerning the extinct people who 
once inhabited Tern, and who have left behind them many relics of a 
wonderful civilization. The M Tnvetf' furnish valuable information 
concerning the country and Itl inhabitants as they now are. Illustrated. 

Ancient Monasteries of the East— Curzon ■ 1 50 

The exploration of those ancient prats of learning has thrown much 
light, upon the researches of khfl historian, the philologist, and the theo- 
logian, as well as the general student of antiquity. Illustrated. 

Discoveries in Babylon & Nineveh— Layard 1 75 

Valuable, alike for the information Imparted with regard to fchefl* most 
Interesting ruins, and the pleasant adventures and observations of the 
author in regions that to most men seem like Fairyland. Illustrated. 

Egypt and the Holy Land— Spencer ... l 75 

Still another volume of eastern travel. The many ineontrovcrtible 
proofs of Scripture observed by the pains-taking modern traveler are 
worth the price of the book. Illustrated. 

St. Petersburgh— Jermann l oo 

Americans arc less familiar with the history and social customs of the 
Russian people than those of any other modern civilized nation. Oppor- 
tunities such as this book affords are not, therefore, to be neglectod. 

The Polar Regions— Osborn 1 25 

A thrilling and intensely interesting narrative of one of the famous ex- 
peditions in search of Sir John Franklin— unsuccessful in its main object, 
but adding many facts to the repertoire of science. 

Thirteen Months in the Confederate Army 75 

The author, a northern man conscripted into the Confederate service, 
and rising from the ranks by soldierly conduct to positions of responsi- 
bility, had remarkable opportunities for the acquisition of facts respect- 
ing the conduct of the Southern armies, and the policy and deeds of tbeir 
leaders. He participated in many engagements, and his book is one of 
the most exciting narratives of advntir.v ev.'r published. Mr. Steven- 
son takes no ground as a partisan, but views the whole subject as with the 
eye of a neutral — only interested in subserving the ends of history by the 
contribution of impartial facts. Illustrated. 

29 



National School Zibrary. 



LIBRARY OF HISTORY. 
History of Europe— Alison $2 50 

A reliable and standard work, which, covers with clear, connected, and 
complete narrative, the eventful occurrences transpiring from A. D. 1789 
to 1815, being mainly a history of the career of Napoleon Bonaparte. 

History of England— Berard l 50 

Combining a history of the social life of the English people with that of 
the civil and military transactions of the realm. 

History of Rome— Ricord l 25 

Possesses all the charm of an attractive romance. The fables with 
which this history abounds are introduced in such away as not to deceive 
the inexperienced reader, while adding vastly to the interest of the work 
and affording a pleasing index to the genius of the Roman people. Illus- 
trated. 

The Republic of America— Willard ... 2 50 
Universal History in Perspective— Willard 2 50 

From these two comparatively brief treatises the intelligent mind may 
obtain a comprehensive knowledge of the history of the world in both 
hemispheres. Mrs. Willard 1 s reputation as an historian is wide as the 
land. Illustrated. 

Ecclesiastical History— Marsh 1 88 

A history of the Church in all ages, with a comprehensive review of all 
forms of religion from the creation of the world. No other source affords, 
in the same compass, the information here conveyed. 

History of the Ancient Hebrews— Mills • • l 75 

The record of " God's people" from the call of Abraham to the destruc- 
tion of Jerusalem ; gathered from sources sacred and profane. 

The Mexican War— Mansfield 1 50 

A history of its origin, and a detailed account of its victories ; with 
official dispatches, the treaty of peace, and valuable tables. Illustrated. 

Early History of Michigan— Sheldon ■ • • l 75 

A work of value and deep interest to the people of the "West. Com- 
piled under the supervision of Hon. Lewis Cass. Embellished with por» 
traits. 

30 



National School Library, 



LIBRARY OF BIOGRAPHY. 
Life of Dr. Sam. Johnson— Boswell • • $2 50 

This work has been before the public for seventy roars, with increasing 
approbation. Boswell is known as M the prince of biographers." 



Henry Clay's Life and Speeches— Mallory 

2 vols 5 00 

This great Amorimn statesman commands the admiration, and his 
character and deed* solicit the study of every pat: I 

Life & Services of General Scott— Mansfield i 75 

The hero of the Mexican war, who was f<>r many years the most promi- 
nent figure in American military circle*, ihonld 1 
■whirl of more recent events than those by which 1 
Illustrated. 

Garibaldi's Autobiography l 50 

The Italian patriot's record of his own life, translated and edited by bin 
friend and admirer. A thrilling narrative of a romantic career. With 
portrait. 

Lives of the Signers— Dwight 1 50 

The memory of the noble men who declared onr country free at the 
peril of their own "lives, fortunes and sacred honor," ihoaJ . 
balmed in every American's heart. 

Life of Sir Joshua Reynolds— Cunningham l 50 

A candid, truthful, and appreciative memoir of the great painter, with 
a compilation of his discourses. The volume is a text-book for artists, as 
well as those who would acquire the rudiments of art. With a portrait. 



Prison Life 



Interesting biographies of celebrated prisoners and martyrs, designed 
especially for the instruction and cultivation of youth. 

3] 



Motional School Library* 



LIBRARY OF NATURAL SCIENCE. 
The Treasury of Knowledge & 10 

A cyclopaedia of ten thousand common things, embracing the widest 
range of subject-matter, illustrated. 

Ganot's Popular Physics X 78 

The elements of natural philosophy for both student and the general 
reader. The original work is celebrated for the magnificent character of 
its illustrations, all of which are literally reproduced here. 

Principles of Chemistry— Porter 1 88 

A work which commends itself to the amateur in science by its ertrezir 
simplicity, and careful avoidance of unnecessary detail. Illus::^ 

Class-Book of Botany— Wood 3 50 

Indispensable as a work of reference. Illustrated. 

The Laws of Health— Jarvis 1 50 

This is not an abstract anatomy, but all its teachings are directed to the 
best methods of preserving health, as inculcated by an intelli_ 
ledge of the structure and needs of the human body. iUustra: 

Vegetable & Animal Physiology— Hamilton l 10 

An exhaustive analysis of the conditions of life in all animate nature. 
Illustrated. 

Elements of Zoology— Chambers l 50 

A mnpic Eke animal kingdom as a portion of external nature. 

Illustrated. 

Astronography— Willard 90 

The elements of astronomv in a compact and readable farm. IIIjs- 
trated. % 

Elements of Geology— Page l 10 

The subject presented in U sting and important 

II. -s: rated. 

Lectures on Natural History— Chadbourne 75 

The subject is tere considered in its relations to intellect, taste, health, 
and religion. 

8 2 



JV % atlo7ial School Library. 



VALUABLE LIBRARY BOOKS. 
The Political Manual— Mansfield $1 25 

Every American youth should he familiar with the principles of the 
government under which he lives, especially as the policy of this country 
will one day call upon him to participate in it, at least to the extent of his 
ballot. 

American Institutions— De Tocqueviile . . l 50 
Democracy in America— De Tocqueviile . . 2 50 

The views of this distinguished foreigner on the genius of our political 
institutions are of unquestionable v.ilu . ling from a standpoint 

whence we seldom have an opportunity to hoar. 

Constitutions of the United Stales • • • • 3 oo 

Contains the Constitution of the General G ovcrnmont, and of the seve- 
ral State Governments, the Declaration <>f Independence, and other im- 
portant documents relating to American history. Indispensable as a 
work of reference. 

Public Economy of the United Stales - • • 2 50 

A full discussion of the relations of the Dal with other na- 

tions, especially the feasibility of a free-trade | 

Grecian and Roman Mythology— Dwight . 2 50 

The presentation, in a systematic form, <>f tlie Fablei of Antiquity, 

affords most entertaining reading, and Is valuable to all 1 1 

mythological allusions so frequent in literature, as well ifsot 

the classics who would peruse intelligently ; Illus- 

trated. 

Modern Philology— Dwight 1 75 

The science of language is here placed, In f a moderate 

volume, within the reach of alL 

General View of the Fine Arts— Huntington l 76 

The preparation of this work was suggested by the Interested inquiries 
of a group of young people, concerning the prodnd j lea of the 

great masters of art, whose names only were familiar, i nt is 

sufficient index of its character. 

Morals for the Young— Willard . 6° 

A series of moral stories, by one of the most experienced of American 
educators. Illustrated. 

improvement of the Mind— Isaac Watts • • so 

A classical standard. No young person should grow up without having 
perused it. 



The National Series of Standard School -"Books. 



GEOGRAPHY. 



THE 

NATIONAL GEOGRAPHICAL SYSTEM. 



I. Monteith's First Lessons in Geography, % 

II. Monteith's Introduction to the Manual, 66 

III. Monteith's New Manual of Geography, i 00 

IV. Monteith's Physical & Intermediate Geog. 1 75 

V. McNallys System of Geography, • • ■ i 

The only oompfc graphical Instruction. Ill elrtw Utl — 

Is almost iintTsrsal — Its mertta pal I w of the elements of Its p 

laritj arc found in the I Isnea. 



1. PRACTICAL OBJECT TEACHING. The Jnsant scholar Is first twtr 

to a )>icture whence he may derive notions of the shape ■•' Ihe phenomena 

of day mill night, the distribution of land sod rater, and th« g il dtrtstons, 

which men words would fall entirely to convey Is the untutored sriad. Othei 
tsrei follow on the nunc plan, sad lbs child's mind is called Ides 

without the aid of a pictorial illustration. Oarried on be the higher boons, thin system 

culminates in No. 4. where such matt, is si eUl i" winds, i 

liurities of the BSfth'l crust, clouds and rain, arc | 

apparent to the most ODt— The illustrations used for this pu i posi belong to the 
highest grade of art. 

2. CLEAR, BEAUTIFUL, AND CORRECT MAPS. In the lower numhsn 
the maps avoid tinned '.. whOc lespeetlfely pr nd affording the 

pupil new matter for acquisition each time he approaches in the constantly enlarging 
cnole the point of coincidence with previous I ossein In the more elementary hooks. 
In No. 4, the maps embrace many new and striking features. One of the I 
effective of these is the new plan for displaying on each map the relative siz> 
countries not represented, thus obviating much confusion which has arisen from the 
necessity of presenting maps in the same atlas drawn on different scales. The mnps 
of No. 5 have long been celebrated for their superior beauty and completeness. This 
is the only 6chool-book in which the attempt to make a cmuplete atlas also clear and 
distinct, lias been successful. The map coloring throughout the series is also BO 
able. Delicate and subdued tints take the place of the startling glare of inharmonious 
colors which too frequently in such treatises dazzle the eyes, distract the attention, 
and serve to overwhelm the names of towns and the natural featuree of the land6c;: 

8 



77/ e National 8ei % ies of Standard ScJtool-TZooks, 



GEOGRAPHY-Continued. 

3. THE VARIETY OF MAP EXERCISE, Starting each time from a different 
basis, the pupil in many instances approaches the same fact no less than six timea, 
thus indelibly impressing it upon his memory. At the same time this system is not 
allowed to become wearisome — the extent of exercise on each subject being graduated 
by its relative importance or difficulty of acquisition. 

4. THE CHARACTER AND ARRANGEMENT OF THE DESCRIPTIVE 
TEXT, The cream of the science has been carefully culled, unimportant matter re- 
jected, elaboration avoided, and a brief and concise manner of presentation cultivated. 
The orderly consideration of topics has contributed greatly to simplicity. Due atten 
tion is paid to the facts in history and astronomy which are inseparably connected 
with, and important to the proper understanding of geography — and such inily are 
admitted on any terms. In a word, the National System teaches geography as a 
science, pure, simple, and exhaustive. 

5. ALWAYS UP TO THE TIMES, The authors of these books, editorially 
speaking, never sleep. No change occurs in the boundaries of countries, or of coun- 
ties, no new discovery is made, or railroad built, that is not at once noted and re- 
corded, and the next edition of each volume carries to every school-room the new or- 
der of things. 

6. SUPERIOR GRADATION, This is the only series which furnishes an avail- 
able volume for every possible class in graded schools. It is not contemplated that a 
pupil must necessarily go through every volume in succession to attain proficiency. 
On the contrary, tiuo will suffice, hut three are advised ; and if the course will admit, 
the whole series should be pursued. At all events, the books are at hand for selection, 
and every teacher, of every grade, can find among them one exactly suited to his class. 
The best combination for those who wish to abridge the course consists of Nos. 1, 3, 
and 5, or where children are somewhat advanced in other studies when they com- 
mence geography, Nos. 2, 3, and 5. Where but two books are admissible, Nos. 2 and 
4, or Nos. 3 and 5, are recommended. 

7. FORM OF THE VOLUMES AND MECHANICAL EXECUTION. The 
maps and text are no longer unnaturally divorced iu accordance with the time-hon- 
ored practice of making text-books on this subject as inconvenient and expensive as 
possible. On the contrary, all map questions are to be found on the page opposite the 
map itbelf, and each book is complete in one volume. The mechanical execution is 
unrivalled. Paper and printing are everything that could be desired, and the bind- 
ing is— A. S. Barnes and Company's. 



Ripley's Map Drawing $i 25 

This system adopts the circle as its basis, abandoning the processes by 
triangulation, the square, parallels, and meridians, &c, which have been 
proved not feasible or natural in the development of this science. Suc- 
cess seems to indicate that the circle " has it." 

National Outline Maps 

Foi the school-room walls. In preparation. 

9 



